Question 36 of 68intermediate🔧 ApplyLong Answer4 marks

Show that the vectors 2i^−j^+k^2\hat{i} - \hat{j} + \hat{k}, i^−3j^−5k^\hat{i} - 3\hat{j} - 5\hat{k} and 3i^−4j^−4k^3\hat{i} - 4\hat{j} - 4\hat{k} form the vertices of a right angled triangle.

Correct Answer

Let the position vectors of points A, B, and C be:

A=2i^−j^+k^,B=i^−3j^−5k^,C=3i^−4j^−4k^\mathbf{A} = 2\hat{i} - \hat{j} + \hat{k}, \quad \mathbf{B} = \hat{i} - 3\hat{j} - 5\hat{k}, \quad \mathbf{C} = 3\hat{i} - 4\hat{j} - 4\hat{k}

Step 1: Find the vectors forming the sides of triangle ABC

Vector AB = Position vector of B - Position vector of A

=(i^−3j^−5k^)−(2i^−j^+k^)=−i^−2j^−6k^= (\hat{i} - 3\hat{j} - 5\hat{k}) - (2\hat{i} - \hat{j} + \hat{k}) = -\hat{i} - 2\hat{j} - 6\hat{k}

Vector BC = Position vector of C - Position vector of B

=(3i^−4j^−4k^)−(i^−3j^−5k^)=2i^−j^+k^= (3\hat{i} - 4\hat{j} - 4\hat{k}) - (\hat{i} - 3\hat{j} - 5\hat{k}) = 2\hat{i} - \hat{j} + \hat{k}

Vector CA = Position vector of A - Position vector of C

=(2i^−j^+k^)−(3i^−4j^−4k^)=−i^+3j^+5k^= (2\hat{i} - \hat{j} + \hat{k}) - (3\hat{i} - 4\hat{j} - 4\hat{k}) = -\hat{i} + 3\hat{j} + 5\hat{k}

Step 2: Calculate the squares of magnitudes of each side

∣AB∣2=(−1)2+(−2)2+(−6)2=1+4+36=41|AB|^2 = (-1)^2 + (-2)^2 + (-6)^2 = 1 + 4 + 36 = \mathbf{41} ∣BC∣2=(2)2+(−1)2+(1)2=4+1+1=6|BC|^2 = (2)^2 + (-1)^2 + (1)^2 = 4 + 1 + 1 = \mathbf{6} ∣CA∣2=(−1)2+(3)2+(5)2=1+9+25=35|CA|^2 = (-1)^2 + (3)^2 + (5)^2 = 1 + 9 + 25 = \mathbf{35}

Step 3: Verify Pythagoras Theorem

∣BC∣2+∣CA∣2=6+35=41=∣AB∣2|BC|^2 + |CA|^2 = 6 + 35 = \mathbf{41} = |AB|^2

Since ∣BC∣2+∣CA∣2=∣AB∣2|BC|^2 + |CA|^2 = |AB|^2, by the converse of Pythagoras theorem, triangle ABC is a right-angled triangle with the right angle at vertex C and AB as the hypotenuse.

∴ Triangle ABC is a right-angled triangle with the right angle at C and AB as the hypotenuse.

Exercise: EXERCISE 10.3 | Q: 17 | (Page 25)
For More Understanding

Explanation

This question is directly solved in Example 12 of the textbook. The solution demonstrates that three points form a right-angled triangle when the sum of squares of two sides equals the square of the third side (Pythagorean theorem).

The key steps involve finding the displacement vectors between consecutive vertices, calculating their magnitudes squared, and verifying the Pythagorean relationship. The calculation shows:

∣AB∣2=41=6+35=∣BC∣2+∣CA∣2|AB|^2 = 41 = 6 + 35 = |BC|^2 + |CA|^2

confirming the right angle at C.

Solution Steps

  1. Step 1: Calculate displacement vectors AB, BC, and CA using position vectors

  2. Step 2: Find the square of magnitude of each vector

  3. Step 3: Verify that the sum of squares of two sides equals the square of the third side

  4. Step 4: Conclude using converse of Pythagoras theorem

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