Let \( a = i + j + k,\; b = i - j + k\) and c = î - ĵ - k̂ be three vectors. A vector \( v\) in the plane of \( a\) and \( b\) whose projection on \( c | c| \) is \( 1 3...

A vector in the plane of \( a \) and \( b \) is \( v = a + λ b = (1+λ) i + (1-λ) j + (1+λ) k \). Its projection on \( c = i - j - k √(3) \) is \( (1+λ) - (1-λ) - (1+λ) √(3) = λ - 1 √(3) = 1 √(3) λ = 2\) \( v = 3 i - j + 3 k \) Answer: (A)

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Question 20

Mathematics Scalar and Vector Products Hard

Let \(\rm \vec a =\hat i +\hat j +\hat k,\; \vec b =\hat i -\hat j + \hat k\) and c = î - ĵ - k̂ be three vectors. A vector \(\rm \vec v\) in the plane of \(\rm \vec a\) and \(\rm \vec b\) whose projection on \(\rm \frac {\vec c} {|\vec c|}\) is \(\frac 1 {\sqrt 3},\) is

(A) 3î - ĵ + 3k̂

(B) î - 3ĵ + 3k̂

(C) 5î - 2ĵ + 5k̂

(D) 2î - ĵ + 3k̂

✓ Correct Answer: Option A Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

A vector in the plane of \(\vec{a}\) and \(\vec{b}\) is \(\vec{v} = \vec{a} + \lambda\vec{b} = (1+\lambda)\hat{i} + (1-\lambda)\hat{j} + (1+\lambda)\hat{k}\).

Its projection on \(\hat{c} = \frac{\hat{i} - \hat{j} - \hat{k}}{\sqrt{3}}\) is

\(\frac{(1+\lambda) - (1-\lambda) - (1+\lambda)}{\sqrt{3}} = \frac{\lambda - 1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \Rightarrow \lambda = 2\)

\(\vec{v} = 3\hat{i} - \hat{j} + 3\hat{k}\)

Answer: (A)

Question 20

📘 Step-by-Step Solution & Detailed Explanation:

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