Question

If $\overline a = \hat i + \hat j - 2\hat k$, then the value of $|(\overline a \times \hat i) \times \hat j|^2 + |(\overline a \times \hat j) \times \hat i|^2 + |(\overline a \times \hat k) \times \hat i|^2$

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Answer 2

We are asked to find the value of $|(\overline a \times \hat i) \times \hat j|^2 + |(\overline a \times \hat j) \times \hat i|^2 + |(\overline a \times \hat k) \times \hat i|^2$, given $\overline a = \hat i + \hat j - 2\hat k$.

We can use the vector triple product identity: $( \mathbf{u} \times \mathbf{v} ) \times \mathbf{w} = (\mathbf{u} \cdot \mathbf{w})\mathbf{v} - (\mathbf{v} \cdot \mathbf{w})\mathbf{u}$.

Let $\overline a = a_1\hat i + a_2\hat j + a_3\hat k$. Given $\overline a = \hat i + \hat j - 2\hat k$, we have $a_1 = 1$, $a_2 = 1$, and $a_3 = -2$.

Let's evaluate each term:

Term 1: $|(\overline a \times \hat i) \times \hat j|^2$ Using the identity with $\mathbf{u} = \overline a$, $\mathbf{v} = \hat i$, and $\mathbf{w} = \hat j$: $(\overline a \times \hat i) \times \hat j = (\overline a \cdot \hat j)\hat i - (\hat i \cdot \hat j)\overline a$ Since $\hat i \cdot \hat j = 0$, this simplifies to: $(\overline a \times \hat i) \times \hat j = (\overline a \cdot \hat j)\hat i$ The dot product $\overline a \cdot \hat j = (a_1\hat i + a_2\hat j + a_3\hat k) \cdot \hat j = a_2$. So, $(\overline a \times \hat i) \times \hat j = a_2\hat i$. The magnitude squared is $|a_2\hat i|^2 = a_2^2 |\hat i|^2 = a_2^2 (1)^2 = a_2^2$. For $\overline a = \hat i + \hat j - 2\hat k$, $a_2 = 1$, so the first term is $1^2 = 1$.

Term 2: $|(\overline a \times \hat j) \times \hat i|^2$ Using the identity with $\mathbf{u} = \overline a$, $\mathbf{v} = \hat j$, and $\mathbf{w} = \hat i$: $(\overline a \times \hat j) \times \hat i = (\overline a \cdot \hat i)\hat j - (\hat j \cdot \hat i)\overline a$ Since $\hat j \cdot \hat i = 0$, this simplifies to: $(\overline a \times \hat j) \times \hat i = (\overline a \cdot \hat i)\hat j$ The dot product $\overline a \cdot \hat i = (a_1\hat i + a_2\hat j + a_3\hat k) \cdot \hat i = a_1$. So, $(\overline a \times \hat j) \times \hat i = a_1\hat j$. The magnitude squared is $|a_1\hat j|^2 = a_1^2 |\hat j|^2 = a_1^2 (1)^2 = a_1^2$. For $\overline a = \hat i + \hat j - 2\hat k$, $a_1 = 1$, so the second term is $1^2 = 1$.

Term 3: $|(\overline a \times \hat k) \times \hat i|^2$ Using the identity with $\mathbf{u} = \overline a$, $\mathbf{v} = \hat k$, and $\mathbf{w} = \hat i$: $(\overline a \times \hat k) \times \hat i = (\overline a \cdot \hat i)\hat k - (\hat k \cdot \hat i)\overline a$ Since $\hat k \cdot \hat i = 0$, this simplifies to: $(\overline a \times \hat k) \times \hat i = (\overline a \cdot \hat i)\hat k$ The dot product $\overline a \cdot \hat i = a_1$. So, $(\overline a \times \hat k) \times \hat i = a_1\hat k$. The magnitude squared is $|a_1\hat k|^2 = a_1^2 |\hat k|^2 = a_1^2 (1)^2 = a_1^2$. For $\overline a = \hat i + \hat j - 2\hat k$, $a_1 = 1$, so the third term is $1^2 = 1$.

Summation The total value is the sum of these three terms: Value = $|(\overline a \times \hat i) \times \hat j|^2 + |(\overline a \times \hat j) \times \hat i|^2 + |(\overline a \times \hat k) \times \hat i|^2$ Value = $a_2^2 + a_1^2 + a_1^2$ Substituting the values $a_1 = 1$ and $a_2 = 1$: Value = $1^2 + 1^2 + 1^2 = 1 + 1 + 1 = 3$.

Alternatively, we can compute the cross products directly: $\overline a = \hat i + \hat j - 2\hat k$

1. $\overline a \times \hat i = (\hat i + \hat j - 2\hat k) \times \hat i = (\hat j \times \hat i) - 2(\hat k \times \hat i) = -\hat k - 2\hat j$. $(\overline a \times \hat i) \times \hat j = (-2\hat j - \hat k) \times \hat j = -2(\hat j \times \hat j) - (\hat k \times \hat j) = \overline 0 - (-\hat i) = \hat i$. $|(\overline a \times \hat i) \times \hat j|^2 = |\hat i|^2 = 1$.

2. $\overline a \times \hat j = (\hat i + \hat j - 2\hat k) \times \hat j = (\hat i \times \hat j) - 2(\hat k \times \hat j) = \hat k - 2(-\hat i) = 2\hat i + \hat k$. $(\overline a \times \hat j) \times \hat i = (2\hat i + \hat k) \times \hat i = 2(\hat i \times \hat i) + (\hat k \times \hat i) = \overline 0 + \hat j = \hat j$. $|(\overline a \times \hat j) \times \hat i|^2 = |\hat j|^2 = 1$.

3. $\overline a \times \hat k = (\hat i + \hat j - 2\hat k) \times \hat k = (\hat i \times \hat k) + (\hat j \times \hat k) = -\hat j + \hat i$. $(\overline a \times \hat k) \times \hat i = (\hat i - \hat j) \times \hat i = (\hat i \times \hat i) - (\hat j \times \hat i) = \overline 0 - (-\hat k) = \hat k$. $|(\overline a \times \hat k) \times \hat i|^2 = |\hat k|^2 = 1$.

Summing the magnitudes squared: $1 + 1 + 1 = 3$.

The final answer is $\boxed{3}$.