Question

If \( \vec{a} = 2\hat{i} + \hat{j} + 2\hat{k} \), then the value of \( |\hat{i} \times (\vec{a} \times \hat{i})| + |\hat{j} \times (\vec{a} \times \hat{j})| + |\hat{k} \times (\vec{a} \times \hat{k})|^2 \) is equal to:}

• A. 17
• B. 18
• C. 19
• D. 20

Hint:

Apply the properties of vector products systematically to simplify expressions involving cross and dot products.

Answer 1

The Correct Option is B

Calculating individual terms, and summing them gives the result 18. Detailed steps for each term are omitted for brevity. Using the vector identities, the calculations are as follows: \[ \hat{i} \times (\hat{a} \times \hat{i}) = (\hat{i} \cdot \hat{i})\hat{a} - (\hat{i} \cdot \hat{a})\hat{i} = \hat{j} + 2\hat{k} \] Similarly, for other unit vectors: \[ \hat{j} \times (\hat{a} \times \hat{j}) = 2\hat{i} + 2\hat{k} \] \[ \hat{k} \times (\hat{a} \times \hat{k}) = 2\hat{i} + \hat{j} \] The magnitudes of the resulting vectors are computed as follows: \[ \left\| \hat{j} + 2\hat{k} \right\|^2 = \left\| \hat{j} \right\|^2 + 2\left\| 2\hat{k} \right\|^2 = 1 + 4 \times 2 = 9 \] \[ \left\| 2\hat{i} + 2\hat{k} \right\|^2 = 2^2 + 2^2 = 4 + 4 = 8 \] \[ \left\| 2\hat{i} + \hat{j} \right\|^2 = 2^2 + 1^2 = 4 + 1 = 5 \] Summing these magnitudes: \[ 5 + 8 + 5 = 18 \]

Answer 2

Step 1: Calculating the Cross Product Using Vector Identities
We begin by applying the vector triple product identity to each term in the given problem. The vector identity is: \[ \hat{i} \times (\hat{a} \times \hat{i}) = (\hat{i} \cdot \hat{i})\hat{a} - (\hat{i} \cdot \hat{a})\hat{i} \] Substituting the values into this identity for \( \hat{i} \), we get: \[ \hat{i} \times (\hat{a} \times \hat{i}) = \hat{j} + 2\hat{k} \]
Similarly, applying the identity for \( \hat{j} \) and \( \hat{k} \):
\[ \hat{j} \times (\hat{a} \times \hat{j}) = 2\hat{i} + 2\hat{k} \]
\[ \hat{k} \times (\hat{a} \times \hat{k}) = 2\hat{i} + \hat{j} \]

Step 2: Calculating the Magnitudes of the Resulting Vectors
Now, we calculate the magnitudes of each of the resulting vectors:
1. For \( \hat{j} + 2\hat{k} \):
\[ \left\| \hat{j} + 2\hat{k} \right\|^2 = \left\| \hat{j} \right\|^2 + 2 \left\| 2\hat{k} \right\|^2 = 1 + 2 \times 2 = 5 \] 2. For \( 2\hat{i} + 2\hat{k} \):
\[ \left\| 2\hat{i} + 2\hat{k} \right\|^2 = 2^2 + 2^2 = 4 + 4 = 8 \] 3. For \( 2\hat{i} + \hat{j} \):
\[ \left\| 2\hat{i} + \hat{j} \right\|^2 = 2^2 + 1^2 = 4 + 1 = 5 \]

Step 3: Summing the Magnitudes
Finally, we sum the magnitudes of the three resulting vectors:
\[ 5 + 8 + 5 = 18 \]