Using Dot product to find angles and lengths

Question

I'm having problems with finding what is being asked for. I have been told that using the dot product of vectors would allow me to find the length for $AE$ as well as the angles needed but I just can't seem to work it out

Answer 1

a) Let $u = \overset{\to}{AB}$, $v = \overset{\to}{AD}$, and $w = \overset{\to}{AE}$. Let $\theta$ be the angle between vectors $\mathbf{u}$ and $\mathbf{w}$ (and between $\mathbf{v}$ and $\mathbf{w}$). We have $$ \cos \theta = \dfrac{\mathbf{u} \cdot \mathbf{w}}{|\mathbf{u}| |\mathbf{w}|} = \dfrac{\mathbf{u} \cdot \mathbf{w}}{5|\mathbf{w}|} \, , \qquad \cos \theta = \dfrac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}| |\mathbf{w}|} = \dfrac{\mathbf{v} \cdot \mathbf{w}}{5|\mathbf{w}|} \, , \qquad $$ so equating these yields $\mathbf{u} \cdot \mathbf{w} = \mathbf{v} \cdot \mathbf{w}$. Since $\mathbf{w} = \mathbf{v} + \overset{\to}{DE}$, $$ \mathbf{v} \cdot \mathbf{w} = \mathbf{v} \cdot (\mathbf{v} + \overset{\to}{DE}) = (\mathbf{v} \cdot \mathbf{v}) + (\mathbf{v} \cdot \overset{\to}{DE}) = |\mathbf{v}|^2 + 0 = 5^2 = 25. $$ Thus, $\mathbf{u} \cdot \mathbf{w} = 25$ also. The dot product is commutative, so we can change the order here to get $\mathbf{w} \cdot \mathbf{u} = 25$ and $\mathbf{w} \cdot \mathbf{v} = 25$.

Answer 2

Length of a vector $u$: $$ \lVert u \rVert = \sqrt{u \cdot u} $$ Angle between two vectors $u$ and $v$: $$ u \cdot v = \lVert u \rVert \, \lVert v \rVert \cos \angle(u,v) $$ For non-zero vectors you solve the above for $\cos \angle(u,v)$ and then tfor the angle.

Answer 3

For the first part of the problem, since you’re not given the height of the pyramid, the values of $\mathbf w\cdot\mathbf u$ and $\mathbf w\cdot\mathbf v$ must be constant. In particular, they have these same values when $E$ lies in the plane of the pyramid’s base. The perpendicularity condition means that in this case $E$ lies somewhere on the line $\overline{CD}$. $\overline{AE}$ bisects $\angle{BAD}$, so $E=C$. The segment $\overline{AC}$ is a diagonal of the pyramid’s base and so $AC=\sqrt2AB=5\sqrt2$ and $m\angle{ACB}=\frac\pi4$, therefore $\mathbf w\cdot\mathbf u=\mathbf w\cdot\mathbf v=5\sqrt2\cdot5\cdot\cos\frac\pi4=25$.

For the second part, you can use a symmetry argument.