If A=i cap+2j cap +3k cap, b = -icap +jcap +4kcap, c=3icap-3jcap-12kcap then find thw angle between vectors a+b+c and cross product of a, b in degrees

Given vectors \(\mathbf{A} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\mathbf{B} = -\hat{i} + \hat{j} + 4\hat{k}\), and \(\mathbf{C} = 3\hat{i} - 3\hat{j} - 12\hat{k}\), we need to find the angle between the vector \(\mathbf{A} + \mathbf{B} + \mathbf{C}\) and the cross product \(\mathbf{A} \times \mathbf{B}\).

1. **Find \(\mathbf{A} + \mathbf{B} + \mathbf{C}\):**

\[
\mathbf{A} + \mathbf{B} + \mathbf{C} = (\hat{i} + 2\hat{j} + 3\hat{k}) + (-\hat{i} + \hat{j} + 4\hat{k}) + (3\hat{i} - 3\hat{j} - 12\hat{k})
\]

\[
= (\hat{i} - \hat{i} + 3\hat{i}) + (2\hat{j} + \hat{j} - 3\hat{j}) + (3\hat{k} + 4\hat{k} - 12\hat{k})
\]

\[
= 3\hat{i} + 0\hat{j} - 5\hat{k}
\]

So, \(\mathbf{A} + \mathbf{B} + \mathbf{C} = 3\hat{i} - 5\hat{k}\).

2. **Find the cross product \(\mathbf{A} \times \mathbf{B}\):**

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

\[
= \hat{i} \left(2 \cdot 4 - 3 \cdot 1\right) - \hat{j} \left(1 \cdot 4 - 3 \cdot (-1)\right) + \hat{k} \left(1 \cdot 1 - 2 \cdot (-1)\right)
\]

\[
= \hat{i} (8 - 3) - \hat{j} (4 + 3) + \hat{k} (1 + 2)
\]

\[
= 5\hat{i} - 7\hat{j} + 3\hat{k}
\]

So, \(\mathbf{A} \times \mathbf{B} = 5\hat{i} - 7\hat{j} + 3\hat{k}\).

3. **Find the angle between \(\mathbf{A} + \mathbf{B} + \mathbf{C}\) and \(\mathbf{A} \times \mathbf{B}\):**

Let \(\mathbf{D} = \mathbf{A} + \mathbf{B} + \mathbf{C}\) and \(\mathbf{E} = \mathbf{A} \times \mathbf{B}\).

\[
\mathbf{D} = 3\hat{i} - 5\hat{k}
\]

\[
\mathbf{E} = 5\hat{i} - 7\hat{j} + 3\hat{k}
\]

The angle \(\theta\) between \(\mathbf{D}\) and \(\mathbf{E}\) is given by:

\[
\cos \theta = \frac{\mathbf{D} \cdot \mathbf{E}}{|\mathbf{D}| |\mathbf{E}|}
\]

First, calculate \(\mathbf{D} \cdot \mathbf{E}\):

\[
\mathbf{D} \cdot \mathbf{E} = (3\hat{i} - 5\hat{k}) \cdot (5\hat{i} - 7\hat{j} + 3\hat{k})
\]

\[
= 3 \cdot 5 + 0 \cdot (-7) + (-5) \cdot 3
\]

\[
= 15 - 15
\]

\[
= 0
\]

Since the dot product is zero, the vectors are orthogonal, and thus the angle between them is \(90^\circ\).

**Summary:**

The angle between the vector \(\mathbf{A} + \mathbf{B} + \mathbf{C}\) and the cross product \(\mathbf{A} \times \mathbf{B}\) is \(90^\circ\).

Would you like any further details or have any questions?

Here are 8 related questions to explore further:
1. How do you compute the magnitude of a vector?
2. What are the properties of the cross product of two vectors?
3. How do you find the unit vector of a given vector?
4. What is the geometric interpretation of the dot product?
5. How do you find the area of a parallelogram formed by two vectors?
6. What is the significance of orthogonal vectors in vector spaces?
7. How do you determine if three vectors are coplanar?
8. What is the relationship between the dot product and the angle between two vectors?

**Tip:** Remember that the cross product of two vectors results in a vector that is orthogonal to both original vectors.

Explore how to find the angle between vectors A + B + C and the cross product of vectors A and B. Understand the properties of vectors, cross products, dot products, and their geometrical interpretations. Suitable for advanced high school students studying vectors in mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation