The following diagram shows quadrilateral $ABCD$, with vector $AD$ = vector $BC$, vector $AB$ = (3,1), and vector $AC$ (4,4).
a. Find vector $BC$.
I am not very sure how you can find this vector based on the others. Would it be (3,4)? Using the x in $AB$ and y in $AC$.
b. Show that vector $BD$ = (-2,2).
If we found vector BC, couldn’t you just go left (on the x axis) until you get to -2?
c. Show that vectors $BD$ and $AC$ are perpendicular.
For this, don’t you multiply them together and add and if it equals 0, then it’s perpendicular?
$$ \vec{BC} = \vec{BA}+ \vec{AC}=-\vec{AB}+\vec{AC} \\=-(3,1)+(4,4)=(1,3)$$
$$\vec{BD} = \vec{BA}+\vec{AD}=-\vec{AB}+\vec{BC} \\=-(3,1)+(1,3)=(-2,2)$$
vectors are perpendicular if their dot product equals zero $$ \vec{BD} \cdot \vec{AC} =(-2,2)\cdot(4,4) \\= (-2)(4)+(2)(4)=0 $$