Question

Let → v = 3 → i + 4 → j and → w = 3 → i + 7 → j . Find an exact number c so that → w − c → v is perpendicular to → v . c =

Answer 1

If
`\vec w-c\vec v` is perpendicular to
`\vec v`, then their dot product is zero.

`(\vec w - c \vec v) \cdot \vec v = (\vec w \cdot \vec v) - c(\vec v\cdot \vec v) = (\vec w \cdot \vec v) - c\|\vec v\|^2 = 0`

We have

`\vec v = 3\,\vec\imath + 4\,\vec\jmath \implies \|\vec v\|^2 = 3^2 + 4^2 = 25`

and

`\vec v \cdot \vec w = 9 + 28 = 37`

so that

`37 - 25c = 0`

Solve for
`c`.

`25 c = 37 \implies \boxed{c = (37)/(25)}`