The angle (θ) between two vectors a and b is found with the formula θ = cos -1 [ ( a.Important Points on Angle Between Two Vectors: We will compute the dot product and the magnitudes first: Let a = i + 2 j + 3 k and b = 3 i - 2 j + k. Let us consider an example to find the angle between two vectors in 3D. Thus, the cross-product formula may not be helpful all the time to find the angle between two vectors. Thus, we got two angles and there is no evidence to choose one of them to be the angle between vectors a and b. If we use the calculator to calculate this, θ ≈ 36.87 (or) 180 - 36.87 (as sine is positive in the second quadrant as well). Let us compute the cross product of a and b.īy using the angle between two vectors formula using cross product, θ = sin -1. Then we get:Īngle Between Vectors in 2D Using Cross Product We can either use a calculator to evaluate this directly or we can use the formula cos -1(-x) = 180° - cos -1x and then use the calculator (whenever the dot product is negative using the formula cos -1(-x) = 180° - cos -1x is very helpful as we know that the angle between two vectors always lies between 0° and 180°). Let us compute the dot product and magnitudes of both vectors.īy using the angle between two vectors formula using dot product, θ = cos -1 [ ( a Let us find the angle between vectors using both dot product and cross product and let us see what is the ambiguity that a cross product can cause.Īngle Between Vectors in 2D Using Dot Product Let us consider two vectors in 2D say a = and b =. Let us also see the ambiguity caused by the cross-product formula to find the angle between two vectors. Let us see some examples of finding the angle between two vectors using dot product in both 2D and 3D. Here, sin -1 is read as "sin inverse" and it is called " inverse sine function". This formula causes some ambiguity (which we discuss in the next section) and is not a popular formula to use to find angle between vectors. This is is the formula for the vector angle in terms of the cross product (vector product). Angle Between Two Vectors Using Cross Productīy the definition of cross product, a × b = | a| | b| sin θ \(\hat\) is a unit vector and hence its magnitude is 1. Here, cos -1 is read as "cos inverse" and it is called " inverse cosine function". This is is the formula for the angle between two vectors in terms of the dot product (scalar product). Note that the cross-product formula involves the magnitude in the numerator as well whereas the dot-product formula doesn't.Īngle Between Two Vectors Using Dot Product
b is the dot product and a × b is the cross product of a and b.Angle between two vectors using cross product is, θ = sin -1.Angle between two vectors using dot product is, θ = cos -1 [ ( a.Then here are the formulas to find the angle between them using both dot product and cross product: Let a and b be two vectors and θ be the angle between them. But the most commonly used formula to find the angle between the vectors involves the dot product (let us see what is the problem with the cross product in the next section). Note that PC=PC', for example, since they are the radii of the same circle.)Ī positive angle of rotation turns a figure counterclockwise (CCW),Īnd a negative angle of rotation turns the figure clockwise, (CW).There are two formulas to find the angle between two vectors: one in terms of dot product and the other in terms of the cross product. (The dashed arcs in the diagram below represent the circles, with center P, through each of the triangle's vertices. A rotation is called a rigid transformation or isometry because the image is the same size and shape as the pre-image.Īn object and its rotation are the same shape and size, but the figures may be positioned differently.ĭuring a rotation, every point is moved the exact same degree arc along the circleĭefined by the center of the rotation and the angle of rotation. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation. When working in the coordinate plane, the center of rotation should be stated, and not assumed to be at the origin. A rotation of θ degrees (notation R C,θ ) is a transformation which "turns" a figure about a fixed point, C, called the center of rotation.