Solving for Vector B: A Comprehensive Guide Using Dot and Cross Products

In this article, we will explore how to solve for vector B given the magnitudes and products of vector A and vector B. Specifically, we will examine the given conditions A 3j, A times B 9i, and A cdot B 12. We will use the properties of both the dot product and the cross product to find the components of vector B.

Given Conditions

The problem provides us with three pieces of information:

A 3j A times B 9i A cdot B 12

Let's break down each condition and solve for vector B step-by-step.

Expressing Vector A

Given A 3j, the vector A can be written as:

A 0i 3j 0k (0, 3, 0)

Using the Cross Product

The cross product A times B 9i implies:

A times B 0i 3j 0k will be cross-multiplied with vector B (P, Q, R) to get the result.

Using the formula for the cross product:

det begin{vmatrix} i j k 0 3 0 P Q R end{vmatrix} 9i

Expanding the determinant:

(3R-0)i - (3P-0)j (0-3Q)k 9i 0j - 3Qk 9i

Simplifying the components:

3R 9 implies R 3 -3P 0 implies P 0

So, the B_z component of vector B is 3 and the B_x component is 0.

Using the Dot Product

The dot product A cdot B 12 implies:

0 cdot P 3 cdot Q 0 cdot R 12

Simplifying the equation:

3Q 12 implies Q 4

So, the B_y component of vector B is 4.

Constructing Vector B

Now that we have:

B_x 0 B_y 4 B_z 3

We can write vector B as:

B 0i 4j 3k (0, 4, 3)

Final Answer

Therefore, vector B is:

B 4j 3k

Conclusion

By using the properties of the dot product and the cross product, we were able to determine the components of vector B given the provided conditions.