If A = 3 ax + -4 ay + 5 az determine the y-component of the cross product ax x A :

Given the vectors:

- \( A = 3 \mathbf{a_x} - 4 \mathbf{a_y} + 5 \mathbf{a_z} \)
- \( \mathbf{a_x} \times A \) is the cross product of \( \mathbf{a_x} \) with \( A \).

We need to determine the **y-component** of the cross product.

### Step 1: Use the properties of the cross product.
The cross product of two vectors is calculated using the determinant of a matrix with unit vectors and vector components:

\[
\mathbf{a_x} \times A = \mathbf{a_x} \times (3 \mathbf{a_x} - 4 \mathbf{a_y} + 5 \mathbf{a_z})
\]

### Step 2: Break down the cross product.
Using the distributive property of the cross product, we expand:

\[
\mathbf{a_x} \times A = \mathbf{a_x} \times 3 \mathbf{a_x} - \mathbf{a_x} \times 4 \mathbf{a_y} + \mathbf{a_x} \times 5 \mathbf{a_z}
\]

### Step 3: Simplify each term.
1. \( \mathbf{a_x} \times \mathbf{a_x} = 0 \) (since the cross product of any vector with itself is zero).
2. \( \mathbf{a_x} \times \mathbf{a_y} = \mathbf{a_z} \).
3. \( \mathbf{a_x} \times \mathbf{a_z} = -\mathbf{a_y} \).

Now substitute these into the equation:

\[
\mathbf{a_x} \times A = 0 - 4 \mathbf{a_z} - 5 \mathbf{a_y}
\]

\[
\mathbf{a_x} \times A = -4 \mathbf{a_z} - 5 \mathbf{a_y}
\]

### Step 4: Extract the y-component.
The y-component of this vector is \( -5 \).

Thus, the **y-component** of \( \mathbf{a_x} \times A \) is \( -5 \).

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Would you like further details on this? Here are some related questions to expand on the topic:

1. What is the full vector result of \( \mathbf{a_x} \times A \)?
2. How does the cross product differ from the dot product in terms of geometric interpretation?
3. What are the properties of the cross product that can be applied in 3D vector analysis?
4. How does the result of a cross product change when switching the order of vectors?
5. How can we visualize the direction of a cross product using the right-hand rule?

**Tip:** The cross product is anti-commutative, meaning \( \mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}) \).

If A = 3 ax + -4 ay + 5 az determine the y-component of the cross product ax x A:

Determine the y-component of the cross product ax × A, where A = 3 ax - 4 ay + 5 az. Learn the step-by-step process of vector cross products, including the properties of 3D vectors and the application of the right-hand rule. Suitable for students in Grades 10-12.

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