Question

If for unit vectors $\hat{a}$, $\hat{b}$ and non-zero $\vec{c}$, $\hat{a}\times\hat{b}+\hat{a}=\vec{c}$ and $\hat{b}.\vec{c}=0$, then volume of parallelopiped with coterminous edges $\hat{a}$, $\hat{b}$ and $\vec{c}$ will be (in cu.units)-

• A. -2
• B. -1
• C. 0
• D. 4
• E. 6
• F. 4
• G. 1
• H. $\frac{1}{2}$

Answer 1

Answer: (G)

1

Answer 2

The volume of the parallelopiped with coterminous edges $\hat{a}$, $\hat{b}$ and $\vec{c}$ is given by the magnitude of the scalar triple product: $V = |[\hat{a}, \hat{b}, \vec{c}]| = |\hat{a} \cdot (\hat{b} \times \vec{c})|$.

We are given two conditions:

1. $\hat{a}\times\hat{b}+\hat{a}=\vec{c}$
2. $\hat{b}.\vec{c}=0$

Substitute the first equation into the second equation: $\hat{b} \cdot (\hat{a}\times\hat{b}+\hat{a}) = 0$

Using the distributive property of the dot product: $\hat{b} \cdot (\hat{a}\times\hat{b}) + \hat{b} \cdot \hat{a} = 0$

The term $\hat{b} \cdot (\hat{a}\times\hat{b})$ is a scalar triple product with two identical vectors ($\hat{b}$), so its value is 0. This is because $\hat{a}\times\hat{b}$ is a vector perpendicular to both $\hat{a}$ and $\hat{b}$, so its dot product with $\hat{b}$ is 0.

Thus, the equation simplifies to: $0 + \hat{b} \cdot \hat{a} = 0$ $\hat{b} \cdot \hat{a} = 0$

Since $\hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{a}$, we have $\hat{a} \cdot \hat{b} = 0$.

This implies that the unit vectors $\hat{a}$ and $\hat{b}$ are orthogonal (perpendicular).

Now, let's calculate the volume $V = |\hat{a} \cdot (\hat{b} \times \vec{c})|$.

Substitute $\vec{c} = \hat{a}\times\hat{b}+\hat{a}$ into the expression for the volume: $V = |\hat{a} \cdot (\hat{b} \times (\hat{a}\times\hat{b}+\hat{a}))|$

Using the distributive property of the cross product: $V = |\hat{a} \cdot (\hat{b} \times (\hat{a}\times\hat{b}) + \hat{b} \times \hat{a})|$

Let's evaluate the terms inside the dot product:

The vector triple product $\hat{b} \times (\hat{a}\times\hat{b})$ can be expanded using the formula $\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{A} \cdot \vec{B})\vec{C}$: $\hat{b} \times (\hat{a}\times\hat{b}) = (\hat{b} \cdot \hat{b})\hat{a} - (\hat{b} \cdot \hat{a})\hat{b}$

Since $\hat{b}$ is a unit vector, $\hat{b} \cdot \hat{b} = |\hat{b}|^2 = 1^2 = 1$. We found that $\hat{a} \cdot \hat{b} = 0$. So, $\hat{b} \times (\hat{a}\times\hat{b}) = (1)\hat{a} - (0)\hat{b} = \hat{a}$.

The second term is $\hat{b} \times \hat{a}$. Using the property of cross product, $\hat{b} \times \hat{a} = -(\hat{a} \times \hat{b})$.

Substitute these back into the volume expression: $V = |\hat{a} \cdot (\hat{a} + (-\hat{a} \times \hat{b}))|$ $V = |\hat{a} \cdot (\hat{a} - \hat{a} \times \hat{b})|$

Using the distributive property of the dot product: $V = |\hat{a} \cdot \hat{a} - \hat{a} \cdot (\hat{a} \times \hat{b})|$

Evaluate the terms:

$\hat{a} \cdot \hat{a} = |\hat{a}|^2$. Since $\hat{a}$ is a unit vector, $|\hat{a}| = 1$, so $\hat{a} \cdot \hat{a} = 1^2 = 1$. $\hat{a} \cdot (\hat{a} \times \hat{b})$ is a scalar triple product with two identical vectors ($\hat{a}$), so its value is 0.

Substitute these values back into the expression for $V$: $V = |1 - 0|$ $V = |1|$ $V = 1$

The volume of the parallelopiped is 1 cubic unit.