The projection of u onto v is the vector [tex] \mathrm{proj}_{\bold{v}} \bold{u} = \left( \dfrac{\bold{u} \cdot \bold{v}}{|\bold{v}|^2}\right) \bold{v} [/tex]
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
[tex] \bold{u}_1 = \mathrm{proj}_{\bold{v}} \bold{u} [/tex]
and
[tex] \bold{u}_2 = \bold{u} - \mathrm{proj}_{\bold{v}} \bold{u} [/tex]
Here,
[tex] \bold{u}_1 = \mathrm{proj}_{\bold{v}} \bold{u} = \left( \dfrac{\bold{u} \cdot \bold{v}}{|\bold{v}|^2}\right) \bold{v} \\ \\= \left(\dfrac{\langle -6, 8 \rangle \cdot \langle 7,1 \rangle}{(7)^2 + (1)^2}\right) \langle 7, 1 \rangle \\ \\= \frac{-17}{25} \langle 7, 1 \rangle \\ \\= \langle -4.76,-0.68\rangle [/tex]
and
[tex] \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 [/tex]
so your answer is
⟨-4.76,0.68⟩ + ⟨-1.24, 8.68⟩
