Vector Algebra
Triangles and Vector Properties
Grade 12
Question:
<p>If the vectors \(6\mathbf{i}, 2\mathbf{j}\) and \(3\mathbf{i} + 6\mathbf{j} - 2\mathbf{k}\) form a triangle, determine the type of triangle.</p>
<p>(a) right angled</p>
<p>(b) obtuse angled</p>
<p>(c) equilateral</p>
<p>(d) isosceles</p>
Step-by-Step Solution
Key Concept: Form a triangle from three vectors and compare the magnitudes of the three sides to determine the type.
Step 1: Let \(\mathbf{A} = 6\mathbf{i}\), \(\mathbf{B} = 2\mathbf{j}\), \(\mathbf{C} = 3\mathbf{i} + 6\mathbf{j} - 2\mathbf{k}\) Step 2: Calculate the sides of the triangle using vectors between points. Step 3: \(|AB| = |\mathbf{B} - \mathbf{A}| = |-6\mathbf{i} + 2\mathbf{j}| = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}\) Step 4: \(|BC| = |\mathbf{C} - \mathbf{B}| = |3\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}| = \sqrt{9 + 16 + 4} = \sqrt{29}\) Step 5: \(|CA| = |\mathbf{A} - \mathbf{C}| = |3\mathbf{i} - 6\mathbf{j} + 2\mathbf{k}| = \sqrt{9 + 36 + 4} = \sqrt{49} = 7\) Step 6: Two sides are equal, confirming the triangle is isosceles. ∴ Answer is (d) isosceles
Correct Answer: d