Thus, the normalized vector is:

Apr 21, 2026

\boxed{\begin{pmatrix} \frac{7}{25} \\ \frac{24}{25} \end{pmatrix}}

Apr 21, 2026

A hydrologist models contaminant spread using vectors $\mathbf{p} = \begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}$. Find the projection of $\mathbf{p}$ onto $\mathbf{q}$.

Apr 21, 2026

The projection of $\mathbf{p}$ onto $\mathbf{q}$ is given by:

Apr 21, 2026

\text{proj}_{\mathbf{q}} \mathbf{p} = \frac{\mathbf{p} \cdot \mathbf{q}}{\|\mathbf{q}\|^2} \mathbf{q}

Apr 21, 2026

First, compute the dot product:

Apr 21, 2026

\mathbf{p} \cdot \mathbf{q} = 4(2) + (-1)(0) + 3(-1) = 8 + 0 - 3 = 5

Apr 21, 2026

Next, find $\|\mathbf{q}\|^2$:

Apr 21, 2026

\|\mathbf{q}\|^2 = 2^2 + 0^2 + (-1)^2 = 4 + 0 + 1 = 5

Feb 24, 2026

Now compute the scalar multiplier:

Apr 21, 2026