Science:Math Exam Resources/Courses/MATH152/April 2015/Question A 04

MATH152 April 2015

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Question A 04
Points $A=[2,1],B=[5,7],C=[0,4]$ and $D$ form the vertices of a parallelogram $ABCD$ that is the subject of questions A4 and A5. Note: $B$ and $D$ are points adjacent to $A$. What is the point $D$?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Using the parallelogram law of vector addition, express the vector ${\overrightarrow {AD}}$ in terms of ${\overrightarrow {AB}}$ and ${\overrightarrow {AC}}$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By the parallelogram law of vector addition, when $B$ and $D$ are points adjacent to $A$, we have ${\overrightarrow {AC}}={\overrightarrow {AB}}+{\overrightarrow {AD}},$ so that ${\overrightarrow {AD}}={\overrightarrow {AC}}-{\overrightarrow {AB}}.$ Since $A=[2,1],B=[5,7],C=[0,4]$ are given, we have ${\overrightarrow {AC}}=C-A=[0,4]-[2,1]=[-2,3]$ and ${\overrightarrow {AB}}=B-A=[5,7]-[2,1]=[3,6].$ Plugging these vectors into the equation, we get $D-[2,1]={\overrightarrow {AD}}={\overrightarrow {AC}}-{\overrightarrow {AB}}=[-2,3]-[3,6]=[-5,-3].$ Adding $[2,1]$ to both sides, we finally obtain the point $D$: $D=[-5,-3]+[2,1]=\color {blue}[-3,-2].$