To determine the quadrant of the vector product $\alpha \times \beta$ on the $x$-$y$ plane, let's analyze the given vectors and calculate the cross product.

Given Vectors

1. $\alpha = a \hat{i} - b \hat{j}$
2. $\beta = c \hat{k}$

where $a$, $b$, and $c$ are positive.

Cross Product Calculation

The cross product $\alpha \times \beta$ is calculated as follows: $\alpha \times \beta = (a \hat{i} - b \hat{j}) \times (c \hat{k})$

Applying the distributive property of the cross product: $= a \hat{i} \times c \hat{k} - b \hat{j} \times c \hat{k}$ $= ac (\hat{i} \times \hat{k}) - bc (\hat{j} \times \hat{k})$

Using the right-hand rule for the unit vectors:

• $\hat{i} \times \hat{k} = -\hat{j}$
• $\hat{j} \times \hat{k} = \hat{i}$

Thus: $\alpha \times \beta = ac (-\hat{j}) - bc (\hat{i})$ $= -bc \hat{i} - ac \hat{j}$

Vector Components

The resulting vector $\alpha \times \beta = -bc \hat{i} - ac \hat{j}$ has:

• An $x$-component of $-bc$ (negative)
• A $y$-component of $-ac$ (negative)

Since both components are negative, $\alpha \times \beta$ points into the third quadrant of the $x$-$y$ plane.

Answer

The correct answer is:

• III

Would you like more details or have any questions?

Here are five related questions for further understanding:

1. What is the physical significance of the cross product in three-dimensional space?
2. How does the right-hand rule help determine the direction of the cross product?
3. How would the cross product change if $a$ or $b$ were negative?
4. What happens if both vectors lie in the same plane (e.g., both in the $x$-$y$ plane)?
5. How would you compute the cross product if given vectors in spherical or polar coordinates?

Tip: In vector calculations, always verify each component's sign to determine direction accurately, especially in physics-related problems.