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To find the cross product \(\vec{a} \times \vec{b}\) for vectors \(\vec{a} = \langle -1, -3, -5 \rangle\) and \(\vec{b} = \langle 1, -2, 3 \rangle\), we can use the formula for the cross product of two vectors in three-dimensional space.

### Formula:
If \(\vec{a} = \langle a_1, a_2, a_3 \rangle\) and \(\vec{b} = \langle b_1, b_2, b_3 \rangle\), then:
\[
\vec{a} \times \vec{b} = \langle a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1 \rangle
\]

### Step-by-Step Calculation:

1. **First Component**:
   \[
   a_2 b_3 - a_3 b_2 = (-3)(3) - (-5)(-2) = -9 - 10 = -19
   \]

2. **Second Component**:
   \[
   a_3 b_1 - a_1 b_3 = (-5)(1) - (-1)(3) = -5 + 3 = -2
   \]

3. **Third Component**:
   \[
   a_1 b_2 - a_2 b_1 = (-1)(-2) - (-3)(1) = 2 + 3 = 5
   \]

### Result:
\[
\vec{a} \times \vec{b} = \langle -19, -2, 5 \rangle
\]

So, the cross product \(\vec{a} \times \vec{b}\) is:
\[
\langle -19, -2, 5 \rangle
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you determine the direction of a cross product vector?
2. What is the geometric interpretation of the cross product?
3. How can you verify the result of a cross product?
4. What properties does the cross product have in vector calculus?
5. How is the cross product used in physics, such as in calculating torque?

### Tip:
For vectors \(\vec{a}\) and \(\vec{b}\) in three-dimensional space, the cross product \(\vec{a} \times \vec{b}\) is always perpendicular to both \(\vec{a}\) and \(\vec{b}\).

Let \( \vec{a} = \langle -1, -3, -5 \rangle \) and \( \vec{b} = \langle 1, -2, 3 \rangle \). Find \( \vec{a} \times \vec{b} \).

Learn how to find the cross product of vectors \( \vec{a} = \langle -1, -3, -5 \rangle \) and \( \vec{b} = \langle 1, -2, 3 \rangle \) with a step-by-step solution. This problem covers vector algebra and the cross product formula. Suitable for students in Grades 11-12.

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