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Given B = 2i − 3j + 6k, find: i) the magnitude of B, ii) the unit vector of B.

October 27, 2026Physics

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2 parts

7

\frac{2}{7}\vec{i} - \frac{3}{7}\vec{j} + \frac{6}{7}\vec{k}

8 steps

Given the vector $\vec{B} = 2\vec{i} - 3\vec{j} + 6\vec{k}$ .

Magnitude of $\vec{B}$

The magnitude of a vector $\vec{V} = x\vec{i} + y\vec{j} + z\vec{k}$ is given by the formula $|\vec{V}| = \sqrt{x^2 + y^2 + z^2}$ .
Substitute the components of $\vec{B}$ into the formula.
Here, $x=2$ , $y=-3$ , and $z=6$ .
$|\vec{B}| = \sqrt{(2)^2 + (-3)^2 + (6)^2}$
Calculate the squares.
$|\vec{B}| = \sqrt{4 + 9 + 36}$
Sum the values.
$|\vec{B}| = \sqrt{49}$
Take the square root.
$|\vec{B}| = \boxed{7}$

Unit vector of $\vec{B}$

The unit vector $\hat{B}$ of a vector $\vec{B}$ is given by the formula $\hat{B} = \frac{\vec{B}}{|\vec{B}|}$ .
Substitute the vector $\vec{B}$ and its magnitude (calculated in part i)) into the formula.
$\hat{B} = \frac{2\vec{i} - 3\vec{j} + 6\vec{k}}{7}$
Distribute the denominator.
$\hat{B} = \boxed{\frac{2}{7}\vec{i} - \frac{3}{7}\vec{j} + \frac{6}{7}\vec{k}}$

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