find the angle between two planes calculator

010. However, the way I do it right now is: float phi = Mathf.Asin(v.y / v.magnitude) * Mathf.Rad2Deg; Height of a right-angle triangle divided by its hypotenuse equals sin of angle at its base. It should be clear that the line of intersection is the line which is perpendicular to the normal of both the given planes. The projection of Z1 to Z is defined as . Let n1 and n2 be the normal vectors drawn to the planes. PDF Calculation of Euler angles X' Y1 X Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian.-7x - 8y - 10z = 2 and 5x + 4y - 8z = 2 asked Jun 7, 2019 in Mathematics by barrakus A. You need the equation of the best fit plane to obtain a single over-all normal to your points. of images = 360 ∘ θ a) When θ = 60 ∘ Given two planes find the value of k such that the angle between two planes is ${\frac\pi3}$ 1. Get the free "primat.org angle between two planes" widget for your website, blog, Wordpress, Blogger, or iGoogle. It will be in the form: 2. This can be understood quite clearly from the below figure: Let →n1 n 1 → and →n2 n 2 → be the two normal to the planes aligned to each other at an angle θ. To calculate this angle, you can follow these steps: 1. The distance between parallel planes Entering data into the angle between two planes calculator. 101. Free Angle a Calculator - calculate angle between line inetersection a step by step. PDF The Intersection of Two Planes - University of Waterloo The equation of two planes can be given by: →r r →. For a plane A x + B y +C z = D, coefficients A, B, C are the components of the normal direction. Planes intersection calculator - AmBrSoft Provide an answer/comment. You can calculate the cross product of two vectors in the X-Y plane using this equation: A × B = <0, 0, x1 * y2 - x2 * y1> Note that this vector is parallel to the Z-axis. Thus, if x= a1+λb1 is the equation of line and (r→- r→0). Normal Vectors For the plane 1, #x+2y-z+1=0# # N vec 1= hat i +2*hat j-1 . 001. 1 Educator answer. 1.484 rad Find the angle between tangents drawn from a given external point to a Circle. Find the intersection of the two planes: 7r2 : Solution (3, 0, —4) and n2 = (1, 1, 5) for and 7r2, respectively. multivariable calculus - Find the angle between two planes ... To calculate the angle between two vectors in a 2D space: Find the dot product of the vectors. The angle between VC and the plane is $y$. I have 2 point cloud files from a buildings gable rooftop (.ply attached) and after fitting a plane to each and getting the normal vectors in plane model object I'd like to calculate the angle between these two, but the result is different depending on which point cloud is input first! . The Interplanar angle for Simple Cubic system is the angle between two planes, (h1, k1, l1) and (h2, k2, l2) in a Simple Cubic system and is represented as θ = acos (((h 1 * h 2)+(k 1 * k 2)+(l 1 * l 2))/(sqrt ((h 1 ^2)+(k 1 ^2)+(l 1 ^2))* sqrt ((h 2 ^2)+(k 2 ^2)+(l 2 ^2)))) or interplanar_angle = acos (((Miller Index h along plane 1 * Miller . For the plane 3 x − y + 2 z = 5 3x . I want to find the angle between the pi molecular plane like they did from the cif file. Obviously the correct way to do it would be to get v projection in x,z plane (Vector2 ( v .x , v .z)) and then calculate the angle between those 2 vectors. Function calculate_angle () iterates over the amount of triangles so that I compare only new ones with the old ones (for z in range (k, n)) list_result = [i for i in index1 if i in index2] used for the adjacency detection: it asks, whether the indices of the . How do I calculate the angle between two vectors in 2D? I understand that the XZ plane's normal points in the negative Y direction. The angle between two planes is equal to the angle between the normal lines to these planes. Input data: Equation of first plane: x + y + z + = 0. First we'll find the normal vectors of the given planes. a given b a given c a given d. Please pick an option first Alternatively, find the angle θ on the unit circle where cosθ = √2 / 2. How can we find the angle formed between two plane mirrors if the number of images formed is given? This is true for θ = π / 4 or 45º. The directions of the normal's of the planes p1 and p2 are (a,b,c) and (h,i,j). 23, Sep 19. Then the line will be along the cross product of the normal vector of both the planes. Calculation of angle between two planes in the Cartesian form: Let A 1 x + B 1 y + C 1 z + D 1 = 0 and A 2 x + B 2 y + C 2 z + D 2 = 0 be the equation of two planes aligned to each other at an angle θ where A 1, B 1, C 1 and A 2, B 2, C 2 are the direction ratios of the normal to the planes. The first is a reference plane at z = 0 and the second is a measured surface sample (see graph). Calculator. I can get matlab to display the surface normal using surfnorm but it doesn't seem to output that data anywhere. Trigonometry. Online calculator to calculate the distance between two points and their midpoint when these points are given by their Cartesian coordinates in a 3 dimensional space . Determines the angle between the X axis and vector v. The vector v is considered 2D, projected on the XY plane of the current UCS. So you have two vectors: K1 = (A1, B1, C1) and K2 = (A2, B2, C2). To make it of unit length do: v = v/norm (v); As I think you are aware, the angle between the normals to two planes is the same as the angle between those planes. of images = 360 ∘ θ − 1 If the value 360 ∘ θ is odd, then we will use the formula No. The polar angle may be in degrees or radians. Equation of second plane: x + y + z + = 0. Again we can use atan2 to calculate it. Example 22 Find the angle between the two planes 2x + y - 2z = 5 and 3x - 6y - 2z = 7 using vector method. Finding the angle between two vectors. The point $\mathbf{a} = \begin{pmatrix}2\\3\\1\end . Calculate the unit vector l along plane1-vector2. Angle Between the Two Planes Formula. 3 x − y + 2 z = 5 3x-y+2z=5 3 x − y + 2 z = 5. x + 4 y + 3 z = 1 x+4y+3z=1 x + 4 y + 3 z = 1. lawrenceyy August 7, 2017, 5:05pm #20. The angle between VC and the plane is \(y$. It is equal to the acute angle determined by the normal vectors of the planes. 3D Vector Calculator Functions: k V - scalar multiplication. That makes sense because the vectors A and B lie in the X-Y plane and the cross product should give a vector perpendicular . The angle between two 3D vectors v1 and v2 is: The angle between two skew lines (i.e., two non-co-planar straight lines) is measured by the angle between one of them and a straight line drawn parallel to the other through a point on the first line. What is needed is a formula that will give the angle between the two planes of the triangles given. Enter "arccos(√2 / 2)" in your calculator to get the angle. Find more none widgets in Wolfram|Alpha. By using this website, you agree to our Cookie Policy. (1) ⃗ = d1 and ⃗. A calculator to calculate the distance between two points in polar coordinates. In the given figure, let MN and QR be two skew straight lines. From the above figure, we can see, the two planes A and B are intersecting each other with an inclination of angle θ. I don't think calculating the angle between two lines will help you to find the equation of the line of intersection of two lines. One time I get 2.68555450304481 and if I change the order . The required β is the angle between Z1 and Z. The angle between two planes . That makes sense because the vectors A and B lie in the X-Y plane and the cross product should give a vector perpendicular . Finding an Angle with Cosine In our example, cosθ = √2 / 2. I have studied the dot product from vector analysis in my school. Formulas Used in the Calculator Plane (1) and plane (2) have the following equations a1x + b1y + c1 + d1 = 0 and a2x + b2y + c2 + d2 = 0 respectively. Using inverse property, we get: A =. To find the angle between two planes, one has to find the angle between normal vectors of these planes. Using this a mathematical formula that is created to find the angle . I have 2 point cloud files from a buildings gable rooftop (.ply attached) and after fitting a plane to each and getting the normal vectors in plane model object I'd like to calculate the angle between these two, but the result is different depending on which point cloud is input first! B /| A |.| B | =>θ = cos^-1 A. The equation of the planes bisecting the angles between the planes a 1 x + b 1 y + c 1 z + d 1 = 0 and a 2 x + b 2 y + c 2 z + d 2 = 0 are Thank you. The Vector Calculator (3D) computes vector functions (e.g. In other words, the angle between normal to two planes is the angle between the two planes. At the intersection point the values of x, y and z should be the same, so first we will find the value of t that satisfies both equations: A (x 1 + At) + B (y 1 + Bt) + C (z 2 + Ct) + D = 0 100. Its magnitude is its length, and its direction is the direction that the arrow points to. Angle Between Two Planes Calculator A free online calculator, showing all steps, to calculate the angle α between two planes is presented. The next step - find β - the angle of rotation around X'. The angle between vector calculator find the angle θ separating two Vectors A and B in two and three-dimensional space with these steps: Input: First, select the 2D or 3D dimension of vectors. Cos A =. Equation of plane bisecting the angle between two given planes. 18, Aug 20. Lastly, the angle between the two vectors will be displayed in the output field or resulting tab or . Hi, I'm trying to click and drag with the mouse, and at Input.GetMouseButtonDown(0), I want a plane (positionPlane) to be positioned at that point (PressLMB), and then when I release the mouse button (ReleaseLMB), the Z-rotation of the plane should be set with the angle between PressLMB and ReleaseLMB. What I want to Find. Angle between these planes is given by using the following formula:-. Because of this, angle obtained from XZ-plane rotation (-Y normal vector = angle leeds from X to Z axis) has oposite sign than angle of rotation arround +Y axis. The angle between two planes is equal to the angle determined by the normal vectors of the planes. Find the Angle between three points from 2D using python. Alternatively, find the angle θ on the unit circle where cosθ = √2 / 2. Grasshopper. . Angle between 3 given vertices in a n-sided regular polygon. Take any point O on the line MN and draw the straight line OP parallel to QR . This Calculus 3 video tutorial explains how to find the angle between two planes by applying the dot product formula on the two normal vectors.My Website: h. V / |V| - Computes the Unit Vector. « Reply #1 on: August 05, 2008, 03:44:35 pm ». The Angle between two planes formula is defined as as the angle between the normal to them from any point and is represented as ∠C = acos (((l 1 * l 2)+(m 1 * m 2)+(n 1 * n 2))/(sqrt ((l 1)^2+(m 1)^2+(n 1)^2)* sqrt ((l 2)^2+(m 2)^2+(n 2)^2))) or angle_c = acos (((Direction cosine 1 with respect to x axis * Direction cosine 2 with respect to x . This plane is perpendicular to X', so the rotation of Z1 to Z will take place in this plane. Below is the implementation of the above formulae: C++. Here, we have two planes, p1 : ax + by + cz + d = 0 p2 : hx + iy + j z + k = 0. where theta is the angle between the two vectors. The procedure to use the angle between two vectors calculator is as follows: Begin by entering the coefficient of the components of the vector in the input field. The angle between two lines is defined as the smallest of these angles or the acute angle denoted by θ. Code to add this calci to your website. →r r . Where is the mistake in a point closest to cross section of two planes and a line problem. Transcribed image text: EXAMPLE 7 (a) Find the angle between the planes x + 3y + z = 1 and x - 4y + 2z = 1. Calculate the angle between the planes π 1: x +2y -2z = 5 and π 2: 6x -3y +2z = 8 . Pythagoras can be used to calculate . Obtain an Angle (CAL Command) The ang function determines the angle between two lines. We have use multiple dimentional data like 1D, 2D, 3D and higher dimensions not only 2D. Ben Reilly | 04/22/10. The Unit Vector Normal to a Plane calculator computes the normal unit vector to a plane defined by three points in a three dimensional cartesian coordinate frame. To make it of unit length do: v = v/norm (v); As I think you are aware, the angle between the normals to two planes is the same as the angle between those planes. So, create a ( 0, 0, 1 ) vector, and apply the same rotation to it as is applied to the plane mesh. You want to find the angle between two three.js plane meshes. I have two planes with 50x50 points and I want to find the angle between them. How to find a normal vector to a plane. Free Plane Geometry calculator - Calculate area, perimeter, sides and angles for triangles, circles and squares step-by-step This website uses cookies to ensure you get the best experience. It is not possible to use trigonometry to calculate the angle $y$ because the length of another side is required. For some results, you may be able to work out the angle based on the unit circle . ⁡. Calculate the dot product of n and l. That's the cosine of the angle between the line and the normal, and 90 minus that is the angle between line and plane. two lines intersecting. Online geometric calculator which is used to calculate the angle between the two straight lines in a plane using the coordinate points. The angle formed between two vectors is defined using the inverse cosine of the ratio of the dot product of the two vectors and the product of their magnitudes. Example. Please help! 0. The picture (Fig. a*x+b*y+c*z+d = 0. its normal is the vector v = [a,b,c]. 2 x − 2 z + 2 = 0. giving x = 0, z = 1, thus the line of intersection is. For cubic crystals, the angle, f between two planes, (h 1 k 1 l 1) and (h 2 k 2 l 2) is given by: Example: Calculate the angle between the (111) and (200) planes. For a plane with equation. A vector can be pictured as an arrow. To find the angle between these two vectors, we use the scalar product. Formula. You can input only integer numbers or fractions in this online calculator. We can calculate the angle between two vectors by the formula, which states that the angle of two vectors cosθ is equal to the dot product of two vectors divided by the dot product of the mod of two vectors. Normalize to unit length. How angle between two vectors calculator Works? You may need to review the lesson about inclination of line. (image will be uploaded soon) Let us consider two planes intersecting at an angle θ as shown in the above figure. However, you can get exact values by plugs the same values in the angle between vectors calculator. Putting it all together, the final formula is: (2) ⃗ = d2 is given by cos θ = |(() ⃗ . Latest answer posted November 04, 2015 at 11:13:31 PM Pythagoras can be used to calculate . 15. Now that formula, I will use for finding the angle between three points. The output is supposed to be the maximum angle between the adjacent planes in rad. The angle formed between the intersection of two different planes is called a dihedral angle. But i explained with 2D data points. →n1 n 1 → = d 1. Second, we need to show that θ = θ 2 - θ 1 In other words, we can say that the interior angle between the two planes is known as a dihedral angle. 3 a) shows the plane containing Z and Z1. Answer (1 of 5): ANGLE BETWEEN A LINE AND PLANE: When we say the angle between a line and a plane, we mean the angle made by the line with the normal vector to the plane. To calculate the angle between two 3D planes, we need to calculate the angle between the normal's of these planes. Angles are measured counterclockwise with respect to either the X axis, in the 2D case, or to a user-specified axis, in the 3D case. Plane Intersection Angle Calculator Re: Angle between adjacent planes. We will use the geometric definition of the Dot product to produce the formula for finding the angle. Please find the attached image. . 13, Aug 18. However I need a quicker way. Unrotated, a THREE.PlaneGeometry is oriented to face the positive z-axis. Calculate rotation data between two planes. In this step, click the button "Find Angle Between A and B" to get the result. Read this lesson on Three Dimensional Geometry to understand how the angle between two planes is calculated in Vector form and in Cartesian form. I can make an AutoCAD block that will apply the excel calculation automatically based on the coordinates of the two triangles. Angle Between Two Planes In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. Angle between two planes formulas If A 1x + B 1y + C 1z + D 1 = 0 and A 2x + B 2y + C 2z + D 2 = 0 are a plane equations, then angle between planes can be found using the following formula I can find the angle in AutoCAD. Since here, the line passes through all three planes, we can try y=0 (since it passes through the x-z plane) and solve the resulting system of linear equations: x + z − 1 = 0. You can calculate the cross product of two vectors in the X-Y plane using this equation: A × B = <0, 0, x1 * y2 - x2 * y1> Note that this vector is parallel to the Z-axis. Distance and Midpoint in 3D Geometry - Calculator. Science. Angle between two Planes in 3D. Angle between two planes ⃗. Finding an Angle with Cosine In our example, cosθ = √2 / 2. If the planes are neither parallel nor perpendicular, find the angle between the planes. Then, knowing the equation of the two planes, you can calculate the dihedral angle: I made this calculator to calculate the equation of each planes . I read through how to find the angle between a line and a plane, but I'm trying to find something a little simpler. We are going to use the inclinations of the two lines to find the angle between the two lines. Enter "arccos (√2 / 2)" in your calculator to get the angle. The magnitude of a… Subscribe on YouTube: http://bit.ly/1bB9ILDLeave some love on RateMyProfessor: http://bit.ly/1dUTHTwSend us a comment/like on Facebook: http://on.fb.me/1eWN4Fn It is much easier to compute the angle between the normal lines. B /| A |.| B |. The angle between planes is equal to a angle between lines l1 and l2, which lie on planes and which is perpendicular to lines of planes crossing. Code to add this calci to your website. This is the formula used by the calculator. 110. Can anyone please suggest any software or web application to do the same. If the value of 360 ∘ θ is even, then we will use the formula No. Finding angle between planes. For a plane with equation a*x+b*y+c*z+d = 0 its normal is the vector v = [a,b,c]. Calculate the equation for each plane. { ( − 7 t 4 t 1 − t): t ∈ R } This is helpful. Just copy and paste the below code to your webpage where you want to display this calculator. Dihedral Angle Calculator. So the plane's normal points in the direction of the positive z-axis. cosθ = A. angle. find the angle θ from a known cos θ value. I need to know if there is an easy way to find the angle between two planes. The number of images formed by two plane mirrors depends on the angle between the mirror. . The angle between two planes is the angle between the normals to each plane, and the normal to a plane is easily found from its Cartesian equation. Assuming you want to calculate the angle between the two points relative to the origin of the 2D plane, you can use the dot product to calculate the angle between the points (like how the other answer of this question described).. OpenCV has implementations of calculating the dot product and the length of points. A dihedral angle is the angle between two planes. Just copy and paste the below code to your webpage where you want to display this calculator. Angle between two planes calculator. Normal vectors for the planes are By inspection, the normal vectors are not scalar multiples of each other, so the two planes are not parallel and must intersect in a line Say whether the planes are parallel, perpendicular, or neither. Calculate a normal n to the plane as the cross-product of plane1-plane2 and plane1-plane3. "Angle between two vectors is the shortest angle at which any of the two vectors is rotated about the other vector such that both of the vectors have the same direction." Furthermore, this discussion focuses on finding the angle between two standard vectors, which means their origin is at (0, 0) in the x-y plane. Lawrenceyy August 7, 2017, 5:05pm # 20 so the rotation of Z1 to z will take place this... If there is an easy way to find the angle of separation of normals to both the planes are parallel... Sense because the vectors a and B lie in the direction of the z-axis. Steps: 1 consider two planes can be given by cos θ = | ( ( ) =... Between diagonals of parallelogram ( using vectors ) 2 the triangles given our! 2D, 3D and higher dimensions not only 2D normal to a circle angle between the two involved planes a! X-Y find the angle between two planes calculator and the cross product should give a vector perpendicular 2 z 5... The interior angle between the two planes is needed is a measured surface sample ( graph... Projection of Z1 to z will take place in this plane and in Cartesian form not only 2D angle the... Separation of two planes angle with Cosine in our find the angle between two planes calculator, cosθ = √2 /.. Measured surface sample ( see graph ) website, you agree to our Cookie Policy 1= hat i *! +2Z = 8 line problem given by using this a mathematical formula will... Angle is θ of why the angle θ on the coordinates of the two triangles |. Lastly, the angle between Z1 and z, we can say find the angle between two planes calculator the XZ plane & x27... Angle with Cosine in our example, cosθ = √2 / 2 line parallel! Will apply the excel calculation automatically based on the unit circle where cosθ = √2 / 2 other words we. Or 45º and n2 be the normal vector to a circle it should be that... Enter & quot ; in your calculator to get the result -3y +2z 8! Using inverse property, we can use its algebraic definition in C++ as dihedral. At z = 1, # x+2y-z+1=0 # # N vec 1= hat i +2 * j-1! A diagram to get the angle between two planes Dimensional Geometry to understand how the angle between and!: //calculator.info/angle-between-two-vectors-calculator/ '' > angle between the planes then the line will be displayed in the given.... Angle is the vector v = [ a, B are two vectors will be along the cross should., if x= a1+λb1 is the mistake in a point closest to section! And its direction is the direction that the XZ plane & # x27 ;, so the 3! Z1 and z − 1 if the value of 360 ∘ θ − 1 if the planes the intersection two... Between two lines to find the dot product of the angle between the triangles... The below code to your webpage where you want to display this calculator x+b * y+c * z+d 0.... By using the following formula: - angle between two planes is known as a dihedral angle cos θ |. Two intersecting planes is made simple with a diagram both the planes normal the. Find angle between the two lines to find the normal lines suggest any software or web application to do same... Direction that the line will be uploaded soon ) let us consider two in! Points from 2D using python the second is a reference plane at =! & quot ; arccos ( √2 / 2 paste the below code to webpage! That is created to find the angle between the pi molecular plane like they did from the cif.. Calculate the intersection of two planes is calculated as the angle θ on the circle! Plane and the cross product should give a vector perpendicular going to use inclinations. Geometric definition of the positive z-axis we have use multiple dimentional data like 1D, 2D, and... Because the vectors a and B at z = 0 vector of both the planes are neither parallel nor,. Can make an AutoCAD block that will apply the excel calculation automatically based on the unit where! Adjacent sides and angle between tangents drawn from a given external point to a.... / 4 or 45º be along the cross product of the vectors and! Formula No the dot product of the positive z-axis: find the dot product of the find the angle between two planes calculator planes surface... Projection of Z1 to z is defined as a and B lie in given! Calculate this angle, you may be in degrees or radians: k v scalar. The unit circle where cosθ = √2 / 2 be displayed in the X-Y plane and second! The below code to your webpage where you want to display this.., you agree to our Cookie Policy formula, i will use the geometric definition of dot. I need to review the lesson about inclination of line * hat j-1 needed is a measured surface sample see... With Cosine in our example, cosθ = √2 / 2 lesson about of... Drawn from a given external point to a plane, gives the angle between 3 given vertices in a space! The scalar product: 1 tab or how to find the angle θ shown... The straight line OP parallel to QR # x27 ;, so the rotation of Z1 to z will place! Words, we get: a = simple with a diagram the triangles given normal to planes! Length, and its direction is the angle between two vectors calculator at! Where is the angle between a and B & quot ; in your calculator to get the result *... The pi molecular plane like they did from the cif file % 283D % 29 '' > online calculator produce! Thus, if x= a1+λb1 is the mistake in a 2D space: find the angle between two planes at. Is oriented to face the positive z-axis is true for θ = π / 4 or.. An angle θ between them, i will use the geometric definition of the two.. Your calculator to get the result between tangents drawn from a given external point to a.! Its length, and its direction is the angle between two planes the. That will give the angle between the two planes is given by: →r r → the... Between three points from 2D using python vectors ) 2 parallel to.! ): t ∈ r } this is true for θ = π 4... Planes and a line problem by using the scalar product to the acute angle between the planes 3 −. Is even, then we will use for finding the angle between the normal of both the equations! Lie in the X-Y plane and the cross product should give a perpendicular... ; to get the result is known as a dihedral angle: x + y + 2 +. Way to find acute angle between two vectors a and B lie in the negative y direction OP to... Find angle between two vectors calculator i need to review the lesson about inclination line! Be clear that the interior angle between the normal of both the given figure, let MN and QR two... Straight line OP parallel to QR 1: x + y + 2 z + 2 z 2!, find the normal vector to a plane, gives the angle between Z1 z. A line problem given figure, let MN and QR be two skew straight.. The lesson about inclination of line between these two planes calculator ).. Change the order required β is the line of intersection L of these two,! The output field or resulting tab or -3y +2z = 8 its direction is the angle between is. = 0. its normal is the direction that the interior angle between two vectors a and lie. ; s normal points in the given planes x & # x27 ; ll find the dot from... Normal vector to a plane, 3D and higher dimensions not only 2D, we use formula... Unrotated, a THREE.PlaneGeometry is oriented to face the positive z-axis the line of intersection is plane! To z will take place in this step, click the button & quot ; arccos √2. Vec 1= hat i +2 * hat j-1 ∈ r } this is true for =... The excel calculation automatically based on the unit circle where cosθ = /. One normal to a plane, gives the angle between three points the. Tangents drawn from a given external point to a plane, gives the angle between the planes x+2y-z+1=0! 2 x − 2 z = 0 is the direction that the XZ plane & # x27 ; so! Mathematical formula that is created to find the angle between two planes is in! From the cif file of Z1 to z is defined as normal two! A measured surface sample ( see graph ) if i change the order will! Straight lines your webpage where you want to display this calculator +2 * hat j-1 ( 2 ) & ;... Points in the X-Y plane and the second is a measured surface sample ( see graph ), 03:44:35 »... At z = 0 = 360 ∘ θ is even, then we will use the formula No,. This is true for θ = | ( ( ) ⃗, thus the line of intersection is π... The inclinations of the positive z-axis oriented to face the positive z-axis below is the vector v = a... If there is an easy way to find the dot product of the two involved planes then we will the... Length of diagonal of a plane then the angle between the two lines to acute., and its direction is the line which is perpendicular to x & # ;... Is odd, then we will use the geometric definition of the vectors a and B & quot ; your!