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The given problem involves vectors \( a = \langle 4, -2, 5 \rangle \), \( b = \langle -1, 3, -6 \rangle \), and \( c = \langle 7, -5, 1 \rangle \). We need to compute the cross products:

1. \( a \times b \)
2. \( b \times c \)
3. \( a \times c \)

I'll proceed with the calculations for each of these cross products.

### Cross product formula:
For two vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), the cross product \( \mathbf{u} \times \mathbf{v} \) is given by:
\[
\mathbf{u} \times \mathbf{v} = \langle u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \rangle
\]

Let's compute each one now.The cross products of the vectors are as follows:

1. \( a \times b = \langle -3, 19, 10 \rangle \)
2. \( b \times c = \langle -27, -41, -16 \rangle \)
3. \( a \times c = \langle 23, 31, -6 \rangle \)

Let me know if you'd like more details or explanations on these calculations.

Here are 5 related questions you might explore:
1. What is the geometric interpretation of a cross product between two vectors?
2. How would you compute the magnitude of the cross product?
3. Can two parallel vectors have a non-zero cross product?
4. How do the directions of \( a \times b \) and \( b \times a \) compare?
5. What is the relation between the cross product and the area of the parallelogram formed by two vectors?

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

Given vectors a = <4, -2, 5>, b = <-1, 3, -6>, and c = <7, -5, 1>, find the cross products a × b, b × c, and a × c.

Learn how to compute the cross products of three vectors: a = <4, -2, 5>, b = <-1, 3, -6>, and c = <7, -5, 1>. This detailed solution uses the cross product formula to find a × b, b × c, and a × c, with step-by-step explanations. Suitable for Grades 10-12.

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