Question Type 7: Determining components of vectors that complement another vector Exercises

For $\mathbf{a} = (3, 2, -2)$ and $\mathbf{b} = (1, 0, 1)$, verify that the component of $\mathbf{a}$ perpendicular to $\mathbf{b}$ lies in the plane orthogonal to $\mathbf{b}$ by computing it.

Given $\mathbf{a} = (3, 2, -2)$ and $\mathbf{b} = (1, 0, 1)$, find the vector component of $\mathbf{a}$ perpendicular to $\mathbf{b}$.

Given vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$, determine the component of $\mathbf{a}$ perpendicular to $\mathbf{b}$.

For $\mathbf{a} = (3, -1, 2)$ and $\mathbf{b} = (1, 4, 0)$, determine the vector component of $\mathbf{a}$ parallel to $\mathbf{b}$.

Find the scalar component of vector $\mathbf{a} = \begin{pmatrix} 5 \\ 4 \\ 1 \end{pmatrix}$ in the direction of vector $\mathbf{b} = \begin{pmatrix} 2 \\ 1 \\ 6 \end{pmatrix}$.

Determine the vector projection of $\mathbf{a} = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$ onto $\mathbf{b} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix}$.

Find the vector projection of $\mathbf{b} = \begin{pmatrix} 5 \\ 0 \\ 2 \end{pmatrix}$ onto $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}$.

For vectors $\mathbf{a} = (3, -1, 2)$ and $\mathbf{b} = (1, 4, 0)$, find the scalar component of $\mathbf{a}$ in the direction of $\mathbf{b}$.

Compute the scalar component of $\mathbf{b} = (5, 0, 2)$ in the direction of $\mathbf{a} = (2, -3, 1)$.

Find the vector projection of $\mathbf{a} = \begin{pmatrix} 5 \\ 4 \\ 1 \end{pmatrix}$ onto $\mathbf{b} = \begin{pmatrix} 2 \\ 1 \\ 6 \end{pmatrix}$.

Find the scalar component of the vector $\mathbf{a} = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$ in the direction of the vector $\mathbf{b} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix}$.

Let $\mathbf{a} = \begin{pmatrix} 7 \\ 1 \\ 4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 3 \\ 6 \end{pmatrix}$. Find the vector component of $\mathbf{a}$ perpendicular to $\mathbf{b}$.