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

$\cos(\alpha) = \dfrac{v_1\cdot v_2}{||v_1||\cdot ||v_2||}$

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:

$||v_1|| = \sqrt{2^2+5^2}=\sqrt{29},\quad ||v_1|| = \sqrt{4^2+(-3)^2}=5$

So, we have

$\cos(\alpha) = \dfrac{-7}{5\sqrt{29}} = -0.26$

This implies

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