Question

Given v = 5i - j and w = 3i+5j, find the angle between v and w

Answer 1

To solve this problem, we will use the dot product, recall that:

$u\cdot v=\parallel u\parallel\parallel v\parallel\cos \theta.$

Where u and v are two vectors.

From the above definition, we get that:

$\theta=\cos ^{-1}(\frac{u\cdot v}{\parallel u\parallel\parallel v\parallel}).$

Substituting the given vectors in the formula, we get:

$\theta=\cos ^{-1}(\frac{(5i-j)\cdot(3i+5j)}{\parallel5i-j\parallel\parallel3i+5j\parallel}).$

Now, recall that:

$\begin{gathered} (ai+bj)\cdot(ci+dj)=(a\cdot b)+(c\cdot d)\text{.} \\ \parallel ai+bj\parallel=a^2+b^2. \end{gathered}$

Therefore:

$\theta=\cos ^{-1}(\frac{15-5}{\sqrt{26}+\sqrt{34}})=\cos ^{-1}(\frac{10}{\sqrt[]{26}+\sqrt[]{34}})\text{.}$

Simplifying we get:

$\theta\approx70.3^{\circ}.$

Answer:

$\theta=70.3^{\circ}\text{.}$