Question

If vector 1=(2,5) and vector 2=(4,-3), what is the angle between the two vectors? Round your answer to one decimal place.

Answer 1

The dot product between two vectors is defined as

`v_1\cdot v_2 = ||v_1||\cdot ||v_2||\cdot \cos(\alpha)`

where
`\alpha` is the angle between the two vectors.

So, we deduce

`\cos(\alpha) = (v_1\cdot v_2)/(||v_1||\cdot ||v_2||)`

The dot product is computed as the sum of the product of correspondent coordinates:

`(2,5)\cdot(4,-3) = 2\cdot 4 + 5\cdot(-3) = 8-15 = -7`

whereas the norm of a vector is the square root of the sum of the squares of the coordinates:

`||v_1|| = √(2^2+5^2)=√(29),\quad ||v_1|| = √(4^2+(-3)^2)=5`

So, we have

`\cos(\alpha) = (-7)/(5√(29)) = -0.26`

This implies

`\alpha = \arccos(-0.26) \approx 105`