Question 14 of 68intermediate🔧 ApplyShort Answer2 marks

Show that the vectors 2î - 3ĵ + 4k̂ and -4î + 6ĵ - 8k̂ are collinear.

Correct Answer

Let the given vectors be a⃗=2i^−3j^+4k^\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k} and b⃗=−4i^+6j^−8k^\vec{b} = -4\hat{i} + 6\hat{j} - 8\hat{k}.

We can express vector b⃗\vec{b} as:

b⃗=−4i^+6j^−8k^=−2(2i^−3j^+4k^)=−2a⃗\vec{b} = -4\hat{i} + 6\hat{j} - 8\hat{k} = -2(2\hat{i} - 3\hat{j} + 4\hat{k}) = -2\vec{a}

Since b⃗\vec{b} can be written as a scalar multiple of a⃗\vec{a} (specifically, b⃗=−2a⃗\vec{b} = -2\vec{a}), the two vectors are collinear.

Exercise: EXERCISE 10.2 | Q: 11 | (Page 17)
For More Understanding

Explanation

Two vectors are collinear if and only if one is a scalar multiple of the other. In this case, the second vector (-4î + 6ĵ - 8k̂) is exactly -2 times the first vector (2î - 3ĵ + 4k̂), which proves they lie along the same line. This is a fundamental property used in vector algebra to establish collinearity.

Solution Steps

  1. Step 1: Let a⃗=2i^−3j^+4k^\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k} and b⃗=−4i^+6j^−8k^\vec{b} = -4\hat{i} + 6\hat{j} - 8\hat{k}

  2. Step 2: Factor out -2 from b⃗\vec{b}: b⃗=−2(2i^−3j^+4k^)=−2a⃗\vec{b} = -2(2\hat{i} - 3\hat{j} + 4\hat{k}) = -2\vec{a}

  3. Step 3: Since b⃗=−2a⃗\vec{b} = -2\vec{a} (a scalar multiple), the vectors are collinear.

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