Mathematics

Problem

A vector

\mathbf{r}=k\,\hat{i}+4\,\hat{j}-3\,\hat{k}

is known to have magnitude

9

. (i) Find the value(s) of

k

. (ii) For each value of

k

obtained, write the corresponding unit vector in the direction of

\mathbf{r}

. (iii) Verify that each unit vector you have written indeed has magnitude

1

Intuitive explanation

The magnitude of a vector is found by taking the square root of the sum of squares of its components—this is just the 3D distance formula. Since you know the magnitude must equal 9, you can set up an equation and solve for the unknown component $k$ . Once you have $k$ , creating a unit vector is straightforward: divide every component by the magnitude. Finally, verifying your answer means checking that your algebra was correct by computing the magnitude of what you wrote.

Intuitive explanation

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