https://file.aiquickdraw.com/mathbot/file/17273494732501lrxfgx2.jpg

The task in the image is asking to determine the dot product (hasil kali skalar), the cross product (hasil kali vektor), and the angle between two vectors \( \mathbf{a} \) and \( \mathbf{b} \), given by:

\[
\mathbf{a} = 2\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}
\]
\[
\mathbf{b} = \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}
\]

Let's go through each part:

### 1. Dot Product (Hasil Kali Skalar)

The dot product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is calculated as:

\[
\mathbf{a} \cdot \mathbf{b} = (2)(1) + (4)(3) + (-3)(2)
\]

Calculating this:

\[
\mathbf{a} \cdot \mathbf{b} = 2 + 12 - 6 = 8
\]

So, the dot product \( \mathbf{a} \cdot \mathbf{b} \) is 8.

### 2. Cross Product (Hasil Kali Vektor)

The cross product of \(\mathbf{a}\) and \(\mathbf{b}\) is calculated using the determinant of the following matrix:

\[
\mathbf{a} \times \mathbf{b} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
2 & 4 & -3 \\
1 & 3 & 2
\end{vmatrix}
\]

Expanding this determinant:

\[
\mathbf{a} \times \mathbf{b} = \mathbf{i}[(4)(2) - (-3)(3)] - \mathbf{j}[(2)(2) - (-3)(1)] + \mathbf{k}[(2)(3) - (4)(1)]
\]

Simplifying:

\[
\mathbf{a} \times \mathbf{b} = \mathbf{i}[8 + 9] - \mathbf{j}[4 + 3] + \mathbf{k}[6 - 4]
\]

\[
\mathbf{a} \times \mathbf{b} = 17\mathbf{i} - 7\mathbf{j} + 2\mathbf{k}
\]

So, the cross product \( \mathbf{a} \times \mathbf{b} \) is \( 17\mathbf{i} - 7\mathbf{j} + 2\mathbf{k} \).

### 3. Angle Between the Vectors (Sudut)

The angle \(\theta\) between the two vectors can be found using the formula:

\[
\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}
\]

First, let's calculate the magnitudes \( |\mathbf{a}| \) and \( |\mathbf{b}| \):

\[
|\mathbf{a}| = \sqrt{2^2 + 4^2 + (-3)^2} = \sqrt{4 + 16 + 9} = \sqrt{29}
\]

\[
|\mathbf{b}| = \sqrt{1^2 + 3^2 + 2^2} = \sqrt{1 + 9 + 4} = \sqrt{14}
\]

Now, substitute the values into the cosine formula:

\[
\cos \theta = \frac{8}{\sqrt{29} \times \sqrt{14}} = \frac{8}{\sqrt{406}}
\]

To find the angle \(\theta\), take the arccosine of the result:

\[
\theta = \cos^{-1}\left(\frac{8}{\sqrt{406}}\right)
\]

This gives the angle \(\theta\) between the vectors \( \mathbf{a} \) and \( \mathbf{b} \).

---

Would you like a detailed explanation of any part of this solution? Also, here are 5 related questions that you might find interesting:

1. How do you calculate the vector projection of one vector onto another?
2. What is the geometric significance of the cross product of two vectors?
3. How can the cross product be used to determine the area of a parallelogram formed by two vectors?
4. What does the dot product tell us about the angle between two vectors?
5. How can vector operations be extended to higher dimensions?

**Tip:** When working with vector operations, always ensure your calculations are consistent with the rules of linear algebra, especially when dealing with multiple dimensions.

Jika a = 2i + 4j - 3k dan b = i + 3j + 2k, tentukanlah hasil kali skalar dan hasil kali vektor dan sudut antara kedua vektor yang diketahui ini.

Learn how to calculate the dot product, cross product, and the angle between the vectors a = 2i + 4j - 3k and b = i + 3j + 2k. This detailed step-by-step solution covers vector operations, including key formulas and theorems. Suitable for high school and early college students.

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