The problem in the image asks for the value of $\alpha$ such that the vector $\vec{p}(\alpha, -1, 0)$ is perpendicular to the vector $\vec{q}(2, 4, -2)$.

To solve this problem, we use the condition for perpendicularity of vectors: their dot product must equal zero.

Step 1: Write the dot product formula

The dot product of two vectors $\vec{p} = (p_1, p_2, p_3)$ and $\vec{q} = (q_1, q_2, q_3)$ is: $\vec{p} \cdot \vec{q} = p_1q_1 + p_2q_2 + p_3q_3$ Here:

• $\vec{p} = (\alpha, -1, 0)$
• $\vec{q} = (2, 4, -2)$

Step 2: Calculate the dot product

Substitute the coordinates of the vectors into the formula: $\vec{p} \cdot \vec{q} = \alpha \cdot 2 + (-1) \cdot 4 + 0 \cdot (-2)$ Simplify: $\vec{p} \cdot \vec{q} = 2\alpha - 4$

Step 3: Use the perpendicularity condition

Since the vectors are perpendicular, the dot product equals zero: $2\alpha - 4 = 0$

Step 4: Solve for $\alpha$

$2\alpha = 4$ $\alpha = 2$

Final Answer:

The value of $\alpha$ is 2.

Would you like me to explain further steps or vector operations? Here are some related questions:

1. What is the general condition for vectors to be perpendicular in 3D space?
2. How is the dot product of vectors defined geometrically?
3. If vectors are not perpendicular, how can you calculate the angle between them?
4. What is the cross product of two vectors, and when is it used?
5. How do you find the projection of one vector onto another?

Tip: The dot product of vectors is fundamental for finding angles and checking perpendicularity in both 2D and 3D geometry.