Question

The vector$\overrightarrow{\mathbf{A}}$ has x, y, and z components of 8.00, 12.0, and 4.00 units, respectively. (a) Write a vector expression for $\overrightarrow{\mathbf{A}}$in unit-vector notation. (b) Obtain a unit-vector expression for a vectorrole="math" localid="1663646691086" $\overrightarrow{\mathbf{B}}$ one-fourth the length of$\overrightarrow{\mathbf{A}}$ pointing in the same direction as $\overrightarrow{\mathbf{A}}$. (c) Obtain a unit-vector expression for a vectorrole="math" localid="1663646702975" $\overrightarrow{\mathbf{C}}$ three times the length of$\overrightarrow{\mathbf{A}}$ pointing in the direction opposite the direction of$\overrightarrow{\mathbf{A}}$ .

Short answer

(a) $8\hat{\mathrm{i}}+12\hat{\mathrm{j}}-4\hat{\mathrm{k}}$unit

(b) $2\hat{\mathrm{i}}+3\hat{\mathrm{j}}-\hat{\mathrm{k}}$unit

(c) $-24\hat{\mathrm{i}}-36\hat{\mathrm{j}}+12\hat{\mathrm{k}}$unit

Step-by-step solution

Step 1(a): Determine the vector expression  A→

The vector$\overrightarrow{\mathrm{A}}$has components x, y and z.

The vector$\overrightarrow{\mathrm{A}}$ is the sum of components x, y and z.

Calculate the vector expression$\overrightarrow{\mathrm{A}}$

$\overrightarrow{\mathrm{A}}=\mathrm{x}\hat{\mathrm{i}}+\mathrm{y}\hat{\mathrm{j}}+\mathrm{z}\hat{\mathrm{k}}$

Substitute 8.00 for x, 12.0 for yand -4.00 for z.

$\begin{array}{rcl}\overrightarrow{\mathrm{A}}& =& 8.00\hat{\mathrm{i}}+12.0\hat{\mathrm{j}}+(-4.00)\hat{\mathrm{k}}\mathrm{unit}\\ & =& 8\hat{\mathrm{i}}+12\hat{\mathrm{j}}-4\hat{\mathrm{k}}\mathrm{unit}\end{array}$

Hence,the vector expression $\overrightarrow{\mathrm{A}}$ is $8\hat{\mathrm{i}}+12\hat{\mathrm{j}}-4\hat{\mathrm{k}}$ unit

Step 2(b): Determine the vector expression  B→

The vector$\overrightarrow{\mathrm{B}}$ is a quarter of the vector$\overrightarrow{\mathrm{A}}$ in the same vector's direction$\overrightarrow{\mathrm{A}}$

Determine the vector expression for the given vector$\overrightarrow{\mathrm{B}}$

$\overrightarrow{\mathrm{B}}=\frac{1}{4}\overrightarrow{\mathrm{A}}$

Substitute$8\hat{\mathrm{i}}+12\hat{\mathrm{j}}-4\hat{\mathrm{k}}$for$\overrightarrow{\mathrm{A}}$

$\begin{array}{rcl}\overrightarrow{\mathrm{B}}& =& \frac{1}{4}\times (8\hat{\mathrm{i}}+12\hat{\mathrm{j}}-4\hat{\mathrm{k}})\\ & =& 2\hat{\mathrm{i}}+3\hat{\mathrm{j}}-\hat{\mathrm{k}}\mathrm{unit}\end{array}$

Hence, the vector expression $\overrightarrow{\mathrm{B}}$ is $2\hat{\mathrm{i}}+3\hat{\mathrm{j}}-\hat{\mathrm{k}}$ unit

Step 3(c): Determine the vector expression  C→

The vector$\overrightarrow{\mathrm{C}}$is three times the vector's lengthin the vector's opposite direction$\overrightarrow{\mathrm{A}}$. Calculate the vector expression$\overrightarrow{\mathrm{C}}$

$\overrightarrow{\mathrm{C}}=-3\overrightarrow{\mathrm{A}}$

Substitute$8\hat{\text{i}}+12\hat{\text{j}}-4\hat{\text{k}}$for$\overrightarrow{\mathrm{A}}$

$\begin{array}{rcl}\overrightarrow{\mathrm{C}}& =& -3(8\hat{\mathrm{i}}+12\hat{\mathrm{j}}-4\hat{\mathrm{k}})\\ & =& -24\hat{\mathrm{i}}-36\hat{\mathrm{j}}+12\hat{\mathrm{k}}\mathrm{unit}\end{array}$

Hence, the vector expression $\overrightarrow{\mathrm{C}}$ is$-24\hat{\mathrm{i}}-36\hat{\mathrm{j}}+12\hat{\mathrm{k}}$ unit.