Question

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$, then $\vec{c} \cdot (\vec{a} - 2\vec{b})$ is equal to __________.

Hint:

The cross product of two vectors is perpendicular to both of them. Use this to determine $\vec{c} \cdot \vec{b}$.

Answer 1

Correct Answer: 3

We are given the vector equations $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$. We want to find the value of $\vec{c} \cdot (\vec{a} - 2\vec{b})$.
This can be expanded as:
$$ \vec{c} \cdot (\vec{a} - 2\vec{b}) = \vec{c} \cdot \vec{a} - 2(\vec{c} \cdot \vec{b}) $$
From the given information, $\vec{c} \cdot \vec{a} = \vec{a} \cdot \vec{c} = 3$.
Now we need to find $\vec{c} \cdot \vec{b}$. We know that $\vec{b} = \vec{a} \times \vec{c}$. By the definition of the cross product, the resulting vector $\vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{c}$.
Specifically, the dot product of $\vec{c}$ with $\vec{b}$ must be zero:
$$ \vec{c} \cdot \vec{b} = \vec{c} \cdot (\vec{a} \times \vec{c}) $$
By the properties of the scalar triple product, $\vec{u} \cdot (\vec{v} \times \vec{u}) = [\vec{u} \ \vec{v} \ \vec{u}] = 0$. Since two of the vectors in the triple product are the same, the volume of the parallelepiped they define is zero.
Thus, $\vec{c} \cdot \vec{b} = 0$.
Substituting these values into our expression:
$$ \vec{c} \cdot (\vec{a} - 2\vec{b}) = 3 - 2(0) = 3 $$