...of $\begin{bmatrix}{1} \\ {7}\end{bmatrix}$ onto the line through $\begin{bmatrix}{-4} \\ {2}\end{bmatrix} $ and the origin.
I am not sure how to start this problem. Can someone show me how to do this? Thanks.
$\endgroup$2 Answers
$\begingroup$There are several ways. Here is one.
We can denote any point on the line as $(-4x,2x)$. (This is a multiple of the given point on the line: I changed it to an ordered pair rather than a column vector for easier writing.)
Then you want the vector from $(-4x,2x)$ to $(1,7)$ to be perpendicular to the line from the origin to $(-4x,2x)$. We can show perpendicularity with the dot product, so take the dot product of those two vectors, set it equal to zero, and solve for $x$.
To do this you just need to know how to find a vector from one point to another, and the formula for the dot product given the coordinates of the vectors.
$\endgroup$ $\begingroup$With inner products: the orthogonal projection of vector $e_2$ onto the line directed by vector $e_1$ is $$\frac{\langle e_1,e_2\rangle}{\langle e_1,e_1\rangle}\, e_1$$
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