[Math] Vector line parallel to $x$-axis

linear algebravector-spaces

The points $P$ and $Q$ have position vectors, relative to the origin $O$, given by
$$
\overrightarrow{OP} = 7\mathbf{i} + 7\mathbf{j} – 5\mathbf{k}
\quad\text{and}\quad
\overrightarrow{OQ} = -5\mathbf{i} + \mathbf{j} + \mathbf{k}.
$$
The mid-point of $PQ$ is the point $A$. The plane $\varPi$ is perpendicular to the line $PQ$ and passes through $A$.

  1. Find the equation of $\varPi$, giving your answer in the form $ax+by+cz=d$.
  2. The straight line through $P$ parallel to the $x$-axis meets $\varPi$ at the point $B$. Find the distance $AB$, correct to $3$ significant figures.

I answered the first part, but I don't the second part. Should the $\mathbf{j}$ and $\mathbf{k}$ vectors be zero?

Best Answer

Hint: We know $A$, and the line $AB$ is described by this equation: $$l: A+\lambda\begin{pmatrix}1\\0\\0 \end{pmatrix}$$ Once you have this, you will be able to solve for the point $B$, and with your information on $A$, you should be able to work out the distance.

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