Let's break down the problem step-by-step.

Given:

1. Vector $\vec{a}$: $\vec{a} = 3\hat{i} - 6\hat{j} + x\hat{k}$
2. Vector $\vec{b}$: $\vec{b} = 2\hat{i} - 6\hat{j}$
3. Magnitude of $\vec{a}$: $|\vec{a}| = 9$

To Find:

1. Value of $x$.
2. Magnitude of $|2\vec{a} - \vec{b}|$.

Part (a): Finding $x$

The magnitude of vector $\vec{a}$ is given by: $|\vec{a}| = \sqrt{(3)^2 + (-6)^2 + (x)^2}$

Given that $|\vec{a}| = 9$, we can set up the equation: $9 = \sqrt{3^2 + (-6)^2 + x^2}$

Simplifying inside the square root: $9 = \sqrt{9 + 36 + x^2}$ $9 = \sqrt{45 + x^2}$

Square both sides to remove the square root: $81 = 45 + x^2$

Isolate $x^2$: $81 - 45 = x^2$ $36 = x^2$

Take the square root of both sides: $x = \pm 6$

Part (b): Finding $|2\vec{a} - \vec{b}|$

First, compute $2\vec{a}$: $2\vec{a} = 2(3\hat{i} - 6\hat{j} + x\hat{k}) = 6\hat{i} - 12\hat{j} + 2x\hat{k}$

Now compute $2\vec{a} - \vec{b}$: $2\vec{a} - \vec{b} = (6\hat{i} - 12\hat{j} + 2x\hat{k}) - (2\hat{i} - 6\hat{j})$

Combine like terms: $2\vec{a} - \vec{b} = (6 - 2)\hat{i} + (-12 + 6)\hat{j} + 2x\hat{k}$ $2\vec{a} - \vec{b} = 4\hat{i} - 6\hat{j} + 2x\hat{k}$

The magnitude of this vector is: $|2\vec{a} - \vec{b}| = \sqrt{(4)^2 + (-6)^2 + (2x)^2}$

Substitute $x = \pm 6$: $|2\vec{a} - \vec{b}| = \sqrt{16 + 36 + (2 \times 6)^2}$ $|2\vec{a} - \vec{b}| = \sqrt{16 + 36 + 144}$ $|2\vec{a} - \vec{b}| = \sqrt{196}$ $|2\vec{a} - \vec{b}| = 14$

So, the answers are:

1. Value of $x$: $x = \pm 6$
2. Magnitude of $|2\vec{a} - \vec{b}|$: 14

Would you like any additional details or further explanation?

5 Related Questions:

1. How do you find the magnitude of a vector in 3D space?
2. What is the geometric interpretation of the dot product of two vectors?
3. How can you determine if two vectors are orthogonal?
4. What is the difference between a vector's magnitude and direction?
5. How do you perform scalar multiplication on a vector?

Tip:

When working with vectors, always break them down into their components and use vector operations systematically for clarity and accuracy.