# Thread: solve perpendicular vectors problem

1. ## solve perpendicular vectors problem

(the vectors in the problem are really in column form, but i don't know how to type that so i put them in component form)

vectorP = 2i +5j
vectorQ = -i+ 2j

☆find the value of p·q
(I know the dot product is 8, that was ez) but the next part i don't get...

☆ if... s = kp - (p·q)q
find the value of the constant k such that s is perpendicular to q

plz help

2. The dot product $\mathbf{p}\cdot\mathbf{q}$ increases when $\mathbf{p}$ and $\mathbf{q}$ point in the same direction, and drops to $0$ exactly when $\mathbf{p}$ and $\mathbf{q}$ are perpendicular.

Therefore, $\mathbf{s}$ will be perpendicular to $\mathbf{q}$ when

\begin{aligned}
\mathbf{s}\cdot\mathbf{q}&=(k\mathbf{p}-(\mathbf{p}\cdot\mathbf{q})\mathbf{q})\cdot\mathbf {q}\\
&=k(\mathbf{p}\cdot\mathbf{q})-(\mathbf{p}\cdot\mathbf{q})|\mathbf{q}|^2\\
&=(\mathbf{p}\cdot\mathbf{q})(k-|\mathbf{q}|^2)\\
&=0.
\end{aligned}

Here, we have used the formula $\mathbf{q}\cdot\mathbf{q}=|\mathbf{q}|^2$.

3. Originally Posted by Scott H
The dot product $\mathbf{p}\cdot\mathbf{q}$ increases when $\mathbf{p}$ and $\mathbf{q}$ point in the same direction, and drops to $0$ exactly when $\mathbf{p}$ and $\mathbf{q}$ are perpendicular.

Therefore, $\mathbf{s}$ will be perpendicular to $\mathbf{q}$ when

\begin{aligned}
\mathbf{s}\cdot\mathbf{q}&=(k\mathbf{p}-(\mathbf{p}\cdot\mathbf{q})\mathbf{q})\cdot\mathbf {q}\\
&=k(\mathbf{p}\cdot\mathbf{q})-(\mathbf{p}\cdot\mathbf{q})|\mathbf{q}|^2\\
&=(\mathbf{p}\cdot\mathbf{q})(k-|\mathbf{q}|^2)\\
&=0.
\end{aligned}

Here, we have used the formula $\mathbf{q}\cdot\mathbf{q}=|\mathbf{q}|^2$.
ummm... sorry i feel kinda lame, but im in high school and i kinda got lost in the first step, plz explain, im not so good with the math...i still don't see how to get the value of k

4. Given that

$8(k-|\mathbf{q}|^2)=0,$

we may divide both sides by $8$, giving

$k-|\mathbf{q}|^2=0.$

Now, we may add $|\mathbf{q}|^2$ to both sides, giving

$k=|\mathbf{q}|^2.$

Hope this helps!

5. i get it now, grazie!