Question

Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, projvu

Answer 1

D

Explanation:

edge answer

Answer 2

The projection of u onto v is the vector
`\mathrm{proj}_{\bold{v}} \bold{u} = \left( \frac{\bold{u} \cdot \bold{v}}{|\bold{v}|^2}\right) \bold{v}`

Should we choose to resolve u into components u₁ (parallel to v) and u₂ (orthogonal to v) then we would have u = u₁ + u₂ and

`\bold{u}_1 = \mathrm{proj}_{\bold{v}} \bold{u}`

and

`\bold{u}_2 = \bold{u} - \mathrm{proj}_{\bold{v}} \bold{u}`

Here,

`\bold{u}_1 = \mathrm{proj}_{\bold{v}} \bold{u} = \left( \frac{\bold{u} \cdot \bold{v}}{|\bold{v}|^2}\right) \bold{v} \\ \\= \left((\langle -6, 8 \rangle \cdot \langle 7,1 \rangle)/((7)^2 + (1)^2)\right) \langle 7, 1 \rangle \\ \\= (-17)/(25) \langle 7, 1 \rangle \\ \\= \langle -4.76,-0.68\rangle`

and

`\bold{u}_2 = \bold{u} - \mathrm{proj}_{\bold{v}} \bold{u} \\ \\= \langle -6, 8\rangle - \langle -4.76, -0.68\rangle \\ \\= \langle -1.24, 8.68\rangle`

so your answer is

⟨-4.76,0.68⟩ + ⟨-1.24, 8.68⟩