Projections and orthogonality practice problems
Projections, scalar components, and splitting a vector into parallel and orthogonal parts. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Which pairs of \( \displaystyle \mathbf a = \left\langle -4, 0, 6 \right\rangle \), \( \displaystyle \mathbf b = \left\langle -18, 30, -12 \right\rangle \), \( \displaystyle \mathbf c = \left\langle -6, 1, 3 \right\rangle \) are orthogonal?
Find the scalar projection \( \displaystyle \operatorname{comp}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle 3, -2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle 1, 3 \right\rangle \).
Find \( \displaystyle \operatorname{proj}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle 1, -2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -1, -4 \right\rangle \).
Find \( \displaystyle \operatorname{proj}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle -4, 6 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -5, 1 \right\rangle \).
Find the scalar projection \( \displaystyle \operatorname{comp}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle 3, 2, 0 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -3, -2, -2 \right\rangle \).
Find the scalar projection \( \displaystyle \operatorname{comp}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle 4, 0, -2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle 0, -5, -5 \right\rangle \).
Find \( \displaystyle \operatorname{proj}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle -1, -5 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -1, 2 \right\rangle \).
Write \( \displaystyle \mathbf u = \left\langle -5, -5, -5 \right\rangle \) as the sum of a vector parallel to \( \displaystyle \mathbf v = \left\langle -1, -4, 3 \right\rangle \) and a vector orthogonal to it.
Write \( \displaystyle \mathbf u = \left\langle -4, 5, -6 \right\rangle \) as the sum of a vector parallel to \( \displaystyle \mathbf v = \left\langle -6, -1, 3 \right\rangle \) and a vector orthogonal to it.
Write \( \displaystyle \mathbf u = \left\langle -5, -1 \right\rangle \) as the sum of a vector parallel to \( \displaystyle \mathbf v = \left\langle -1, 5 \right\rangle \) and a vector orthogonal to it.