# solve perpendicular vectors problem

• Dec 7th 2009, 04:34 PM
3k1yp2
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
• Dec 7th 2009, 05:42 PM
Scott H
The dot product $\displaystyle \mathbf{p}\cdot\mathbf{q}$ increases when $\displaystyle \mathbf{p}$ and $\displaystyle \mathbf{q}$ point in the same direction, and drops to $\displaystyle 0$ exactly when $\displaystyle \mathbf{p}$ and $\displaystyle \mathbf{q}$ are perpendicular.

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

\displaystyle \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 $\displaystyle \mathbf{q}\cdot\mathbf{q}=|\mathbf{q}|^2$.
• Dec 7th 2009, 05:50 PM
3k1yp2
Quote:

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

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

\displaystyle \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 $\displaystyle \mathbf{q}\cdot\mathbf{q}=|\mathbf{q}|^2$.

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
• Dec 7th 2009, 06:01 PM
Scott H
Given that

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

we may divide both sides by $\displaystyle 8$, giving

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

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

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

Hope this helps!
• Dec 7th 2009, 06:14 PM
3k1yp2
i get it now, grazie!