Question

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. If \(|\vec{a}| = \sqrt{26}\), \(|\vec{b}| = 7\) and \(|\vec{a} \times \vec{b}| = 35\), then \(|\vec{a} \cdot \vec{b}|\) is equal to	I. 4
B. If \(2\hat{i} - 3\hat{j} + 4\hat{k}\) and \(a\hat{i} - 6\hat{j} + 8\hat{k}\) are collinear, then 'a' is equal to	II. 2
C. If \(2\hat{i} + \hat{j} + \hat{k}\) and \(2\hat{i} - 4\hat{j} + \lambda\hat{k}\) are perpendicular, then \(\lambda\) is equal to	III. 7
D. \((\hat{i} \times \hat{j}) \cdot \hat{k} + (\hat{j} \times \hat{k}) \cdot \hat{i}\) is equal to	IV. 0

Choose the correct answer from the options given below:

• A. A-III, B-II, C-IV, D-I
• B. A-III, B-I, C-IV, D-II
• C. A-II, B-I, C-III, D-IV
• D. A-I, B-IV, C-II, D-III

Hint:

Use \(|\vec{a}\times\vec{b}|^2 + (\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2\), proportional components for collinear and zero dot product for perpendicular.

Answer 1

The Correct Option is B

Step 1: Plan.
We evaluate each item of List-I using vector identities and then match it with List-II.

Step 2: Item A.
We use \(|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2\).
\(35^2 + (\vec{a} \cdot \vec{b})^2 = 26 \times 49\), so \((\vec{a} \cdot \vec{b})^2 = 1274 - 1225 = 49\).
Thus \(|\vec{a} \cdot \vec{b}| = 7\). A matches III.

Step 3: Item B.
Collinear vectors have proportional components: \(\frac{a}{2} = \frac{-6}{-3} = \frac{8}{4} = 2\). So \(a = 4\). B matches I.

Step 4: Item C.
Perpendicular vectors have zero dot product: \(2(2) + 1(-4) + 1(\lambda) = 0\), so \(4 - 4 + \lambda = 0\) and \(\lambda = 0\). C matches IV.

Step 5: Item D.
\(\hat{i} \times \hat{j} = \hat{k}\), so \((\hat{i} \times \hat{j}) \cdot \hat{k} = 1\). Also \(\hat{j} \times \hat{k} = \hat{i}\), so \((\hat{j} \times \hat{k}) \cdot \hat{i} = 1\). The sum is 2. D matches II.

Step 6: Check the options.
The matching is A-III, B-I, C-IV, D-II. Option 1 has B-II and D-I, which is wrong. Option 3 has A-II, wrong. Option 4 has A-I, wrong. Only option 2 is right.

Final Answer:
The correct matching is A-III, B-I, C-IV, D-II, which is option 2.