Mathematics

Problem

Let

\mathbf{p}=x\hat{i}+2\hat{j}-3\hat{k}

and

\mathbf{q}=4\hat{i}-\hat{j}+5\hat{k}

. The resultant vector

\mathbf{r}=\mathbf{p}+\mathbf{q}

satisfies

|\mathbf{r}|=7

. (a) Determine the value(s) of

x

that satisfy this condition. (b) Using the value of

x

obtained, write

\mathbf{r}

in component form. (c) Find the unit vector in the direction of

\mathbf{r}

Intuitive explanation

Vector addition combines components independently, so finding $|\mathbf{r}|$ means squaring and summing the resulting components—then solving for $x$ using the constraint that the magnitude equals 7. Once you have $x$ , the rest follows directly: write out $\mathbf{r}$ in components, then normalize it by dividing by its magnitude to get the unit vector.

Intuitive explanation

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