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Project the vector $v = (4, 1)$ orthogonally onto the line through the origin spanned by $u = (1, 2)$ . What is the second coordinate of the projection? Give a fraction in lowest terms.

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Hint

The projection is a multiple of $u$ , and the multiple comes from $2$ dot products.

Solution

The orthogonal projection of $v$ onto the line is $\frac{v \cdot u}{u \cdot u} \times u$ .
Here $v \cdot u = 4 \times 1 + 1 \times 2 = 6$ and $u \cdot u = 1 \times 1 + 2 \times 2 = 5$ .
So the projection is $\frac{6}{5} \times (1, 2)$ , and its second coordinate is $\frac{6}{5} \times 2 = \frac{12}{5}$ .

Answer: $\frac{12}{5}$

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