We are given two vectors:

$\vec{a} = 2\hat{i} - 6\hat{j} - 3\hat{k}$ and $\vec{b} = x\hat{i} + 2\hat{j} - 4\hat{k}$

The length of the projection of $\vec{a}$ onto $\vec{b}$ is given as $\frac{4}{3}$, and we are asked to determine the value of $x$.

Formula for Projection

The length of the projection of $\vec{a}$ onto $\vec{b}$ is given by:

$\text{Projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

We need to find both the dot product $\vec{a} \cdot \vec{b}$ and the magnitude $|\vec{b}|$.

1. Dot product $\vec{a} \cdot \vec{b}$

The dot product is calculated as:

$\vec{a} \cdot \vec{b} = (2)(x) + (-6)(2) + (-3)(-4)$

Simplifying:

$\vec{a} \cdot \vec{b} = 2x - 12 + 12 = 2x$

2. Magnitude $|\vec{b}|$

The magnitude of $\vec{b}$ is:

$|\vec{b}| = \sqrt{x^2 + 2^2 + (-4)^2} = \sqrt{x^2 + 4 + 16} = \sqrt{x^2 + 20}$

3. Setting up the equation

We know the length of the projection is $\frac{4}{3}$, so we set up the equation:

$\frac{2x}{\sqrt{x^2 + 20}} = \frac{4}{3}$

4. Solving for $x$

Multiply both sides by $\sqrt{x^2 + 20}$:

$2x = \frac{4}{3} \times \sqrt{x^2 + 20}$

Multiply both sides by 3 to eliminate the denominator:

$6x = 4\sqrt{x^2 + 20}$

Now square both sides to remove the square root:

$(6x)^2 = (4\sqrt{x^2 + 20})^2$

$36x^2 = 16(x^2 + 20)$

Expand the right-hand side:

$36x^2 = 16x^2 + 320$

Move all terms to one side:

$36x^2 - 16x^2 = 320$

$20x^2 = 320$

Divide by 20:

$x^2 = 16$

Taking the square root of both sides:

$x = 4 \quad \text{or} \quad x = -4$

Thus, the possible values of $x$ are $4$ or $-4$.

Answer:

The correct option is C: 4 or -4.

Would you like further details on this solution? Here are some related questions:

1. How do you calculate the magnitude of a vector in general?
2. What is the geometric interpretation of a vector projection?
3. Can you explain the dot product's significance in physics and geometry?
4. What is the process for solving quadratic equations like $x^2 = 16$?
5. How do projections relate to work done in physics?

Tip: Always check if the square root in the magnitude calculation leads to a positive result when solving vector projections.