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)((u\cdot v)/(\parallel u\parallel\parallel v\parallel)).`

Substituting the given vectors in the formula, we get:

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

Simplifying we get:

`\theta\approx70.3^(\circ).`

Answer:

`\theta=70.3^(\circ)\text{.}`