Question 13(c) | BetaMath

Question 13(c)

The vector $\underset{\sim}{a}$ is $\displaystyle \begin{pmatrix} 1 \\ 3 \end{pmatrix}$ and the vector $\underset{\sim}{b}$ is $\displaystyle \begin{pmatrix} 2 \\ -1 \end{pmatrix}$

The projection of a vector $\underset{\sim}{x}$ onto the vector $\underset{\sim}{a}$ is $k\underset{\sim}{a}$ , where $k$ is a real number.

The projection of the vector $\underset{\sim}{x}$ onto the vector $\underset{\sim}{b}$ is $p\underset{\sim}{b}$ , where $p$ is a real number.

Find the vector $\underset{\sim}{x}$ in terms of $k$ and $p$ .

Solution

We recall that a vector projection can be expressed via the dot product:

\begin{aligned} \text{Proj}_{\underset{\sim}{a}}{\underset{\sim}{x}} &= \left(|\underset{\sim}{x}|\cos{\theta}\right)\frac{\underset{\sim}{a}}{|\underset{\sim}{a}|} \\ \text{Proj}_{\underset{\sim}{a}}{\underset{\sim}{x}} &= \left(\frac{\underset{\sim}{x}\cdot\underset{\sim}{a}}{|\underset{\sim}{a}|}\right)\frac{\underset{\sim}{a}}{|\underset{\sim}{a}|} \\ \text{Proj}_{\underset{\sim}{a}}{\underset{\sim}{x}} &= \left(\frac{\underset{\sim}{x}\cdot\underset{\sim}{a}}{|\underset{\sim}{a}|^2}\right)\underset{\sim}{a} \\ \text{Proj}_{\underset{\sim}{a}}{\underset{\sim}{x}} &= \left(\frac{\underset{\sim}{x}\cdot\underset{\sim}{a}}{\underset{\sim}{a}\cdot\underset{\sim}{a}}\right)\underset{\sim}{a} \\ \end{aligned}

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