Question

Find the angle between the vectors U=<1,-2> and V=<4,1>.

a.) 167°
b.)13°
c.) 77°
d.) 103°

Answer 1

Determine the vectors you must use to find the angle. Say two vectors OM and OQ intersect at point O, and you need to calculate the angle MOQ. You must use vectors OM and OQ, not MO or QO. If you know MO, multiply it by -1 (negative one) to give OM and use that.

OR

Identify the components of the vector in each direction. If the vector is given as a column vector, the first row usually represents the x-axis, the second row the y-axis, and the third row the z-axis. If the vector is given in the form xi + yj + zk, the coefficients of i,j, and k represent the magnitudes of the components along the x-, y-, and z-axes respectively (i,j, and k are unit vectors along the x-, y-, and z-axes respectively).

Answer 2

OR
Add the three multiplication products together. This is the scalar product of the two vectors. The scalar product, or "dot product", of two vectors is a very useful quantity in geometry and physics. For now, we just use it to aid in the calculation of the angle between two vectors. In a two dimensional vector, the component along the z-axis is zero, so the scalar product is found by considering the components along the x- and y-axes only.

Answer 3

Recall that if θ is the angle between U and V, then:
cosθ = (U • V)/(||U|| * ||V||).

We have:
(a) U • V = <1, -2> • <4, 1> = (1)(4) + (-2)(1) = 4 - 2 = 2
(b) ||U|| = √[(1)^2 + (-2)^2] = √(4 + 1) = √5
(c) ||V|| = √[(4)^2 + 1^2] = √(16 + 1) = √17.

Using the above formula:
cosθ = (U • V)/(||U|| * ||V||) = 2/(√5 * √17) = 2/√85 ==> θ = 77°.