# Math Help - Gr 12. Vector Question Involving Dot Product of Two Vectors

1. ## Gr 12. Vector Question Involving Dot Product of Two Vectors

This is one of the difficult question in this chapter.

The vectors a = 3i - 4j - k (3, -4, -1) and b = 2i + 3j -6k (2, 3, -6) are the diagonals of a parallelogram. Show that this parallelogram is a rhombus, and determine the lengths of the sides and the angles between the sides.

Yes, I know what a rhombus is and some of it's properties.
I used the dot product for the 2 Vectors given which = 0 therefore I know that the 2 vectors bisect each other in the rhombus. So, how do I find the side lengths and the angle? Thanks

2. $s = \sqrt{\left(\frac{|a|}{2}\right)^2 + \left(\frac{|b|}{2}\right)^2}$

$\theta = 2\arctan\left(\frac{|a|}{|b|}\right)$

$\phi = 2\arctan\left(\frac{|b|}{|a|}\right)
$

3. Thanks dude for not giving the answer! After taking a closer look at it I understood how you got the Side equation! But No idea how use arctan since we don't learn that in high school but I used tan instead given that it's right triangles and got the correct angles. Thanks sooo much! Speedy and Fast!

4. Suppose that u & v are two vectors that form the adjacent forming the sides of the parallelogram.
Then we know that $a = u - v\;\& \;b = u + v$ or visa versa.
Thus we have $u = \frac{{a + b}}{2}\;\& \;v = \frac{{a - b}}{2}$. Now use the given to prove that $\left\| u \right\| = \left\| v \right\|$.

5. Thanks Plato for the extra info!