Problem on finding the length of vectors
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Find the length of the 2d vector $2 \ \bfi + 3 \ \bfj$ and the 3d vector $\langle2, 3, 4 \rangle$.
Solution
Recall that
Definition of vector length
The length of the vector $\mathbf{x} = \langle x_1, x_2 \rangle$ is $|\mathbf{x}| = \sqrt{x_1^2 + x_2^2}$.
For the vector $2 \ \bfi + 3 \ \bfj$, we identify that $x_1 = 2$ and $x_2 = 3$.
The length of $2 \ \bfi + 3 \ \bfj$ is $$\left | 2 \ \bfi + 3 \ \bfj \right | = \sqrt{2^2 + 3^2} = \sqrt{13}.$$
Recall that
Definition of vector length
The length of the vector $\mathbf{x} = \langle x_1, x_2, x_3 \rangle$ is $|\mathbf{x}| = \sqrt{x_1^2 + x_2^2 + x_3^2}$.
We get $$ \left | \langle 2, 3, 4 \rangle \right | = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{29}.$$
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