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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.

Diketahui vektor-vektor a = 2i - 6j - 3k dan b = xi + 2j - 4k. Panjang proyeksi vektor a pada b adalah 4/3, maka x = ....

Learn how to solve a vector projection problem where the projection of vector a = 2i - 6j - 3k onto vector b = xi + 2j - 4k has a length of 4/3. Follow the step-by-step solution using vector projection, dot product, and vector magnitude.

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