Sample · Multivariable and Vector Calculus · 1 of 5

Problem

Multivariable and Vector Calculus

Splitting a Vector Along Another

Let $\mathbf a=\langle 1,2,2\rangle$ and $\mathbf b=\langle 3,1,-1\rangle$ .

(a) Compute the scalar projection of $\mathbf b$ onto $\mathbf a$ and the vector projection $\mathrm{proj}_{\mathbf a}\mathbf b$ .

(b) Write $\mathbf b$ as the sum of a vector $\mathbf b_{\parallel}$ parallel to $\mathbf a$ and a vector $\mathbf b_{\perp}$ orthogonal to $\mathbf a$ .

(c) Verify by a direct dot product that $\mathbf b_{\perp}$ is orthogonal to $\mathbf a$ , and verify that $\lvert\mathbf b_{\parallel}\rvert^2+\lvert\mathbf b_{\perp}\rvert^2=\lvert\mathbf b\rvert^2$ . Interpret the last identity geometrically.

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