Show that the vectors a⃗ = 3î − 2ĵ + k̂, b⃗ = î − 3ĵ + 5k̂ and c⃗ = 2î + ĵ − 4k̂ can form the sides of a triangle. Find the length of each side and show that the triangle is right-angled.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Given: a⃗, b⃗, c⃗ as above. To show: they form a triangle that is right-angled. To find: the side lengths.
2 · Write the rule. Triangle law: if one vector is the sum of the other two, the three can be placed head to tail to form a triangle.
3 · Draw the figure. Draw b⃗ from P to Q, then c⃗ from Q to R. The closing side from P to R is b⃗ + c⃗. Label all three arrows.
4 · Add two of the vectors. b⃗ + c⃗ = (1 + 2)î + (−3 + 1)ĵ + (5 − 4)k̂ = 3î − 2ĵ + k̂ = a⃗. So the vectors form a triangle.
5 · Find the lengths.
|a⃗| = √(9 + 4 + 1) = √14
|b⃗| = √(1 + 9 + 25) = √35
|c⃗| = √(4 + 1 + 16) = √21
6 · Test for a right angle. |a⃗|² + |c⃗|² = 14 + 21 = 35 = |b⃗|². By the converse of Pythagoras' theorem, the angle between the sides a⃗ and c⃗ is 90°.
7 · Check and box the answer. a⃗ = b⃗ + c⃗, so they form a triangle with sides √14, √35 and √21. It is right-angled.
Scoring tip: State the triangle law in words before adding. The examiner wants the reason, not just the sum.
