Two vectors are given by a = 4i - 3j fk and b =-;F ;+ 4k. Find (ā) a 7 b, (b) a - b, and (c) a vector c such that a - b + + = 0.

To solve the problem involving the vectors \( \mathbf{a} \) and \( \mathbf{b} \), we first need to clarify the vectors given. It seems there might be a small typo in your question regarding vector \( \mathbf{b} \). I will assume you meant \( \mathbf{b} = -5i + 4k \). If that’s incorrect, please let me know. Now, let’s break down the operations you need to perform step by step.

Vector Definitions

We have the following vectors:

• Vector a: \( \mathbf{a} = 4\mathbf{i} - 3\mathbf{j} + 0\mathbf{k} \)
• Vector b: \( \mathbf{b} = -5\mathbf{i} + 0\mathbf{j} + 4\mathbf{k} \)

Part (a): Finding \( \mathbf{a} + \mathbf{b} \)

To find the sum of the two vectors, we simply add their corresponding components:

• For the \( \mathbf{i} \) component: \( 4 + (-5) = -1 \)
• For the \( \mathbf{j} \) component: \( -3 + 0 = -3 \)
• For the \( \mathbf{k} \) component: \( 0 + 4 = 4 \)

Thus, the result of \( \mathbf{a} + \mathbf{b} \) is:

Result: \( \mathbf{a} + \mathbf{b} = -1\mathbf{i} - 3\mathbf{j} + 4\mathbf{k} \)

Part (b): Finding \( \mathbf{a} - \mathbf{b} \)

Next, we subtract vector \( \mathbf{b} \) from vector \( \mathbf{a} \) by subtracting the corresponding components:

• For the \( \mathbf{i} \) component: \( 4 - (-5) = 4 + 5 = 9 \)
• For the \( \mathbf{j} \) component: \( -3 - 0 = -3 \)
• For the \( \mathbf{k} \) component: \( 0 - 4 = -4 \)

Thus, the result of \( \mathbf{a} - \mathbf{b} \) is:

Result: \( \mathbf{a} - \mathbf{b} = 9\mathbf{i} - 3\mathbf{j} - 4\mathbf{k} \)

Part (c): Finding a vector \( \mathbf{c} \) such that \( \mathbf{a} - \mathbf{b} + \mathbf{c} = 0 \)

To find vector \( \mathbf{c} \), we can rearrange the equation:

\( \mathbf{c} = -(\mathbf{a} - \mathbf{b}) \)

From part (b), we found \( \mathbf{a} - \mathbf{b} = 9\mathbf{i} - 3\mathbf{j} - 4\mathbf{k} \). Therefore:

• For the \( \mathbf{i} \) component: \( -9 \)
• For the \( \mathbf{j} \) component: \( 3 \)
• For the \( \mathbf{k} \) component: \( 4 \)

Thus, the vector \( \mathbf{c} \) is:

Result: \( \mathbf{c} = -9\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} \)

Summary of Results

To summarize, we have:

• Part (a): \( \mathbf{a} + \mathbf{b} = -1\mathbf{i} - 3\mathbf{j} + 4\mathbf{k} \)
• Part (b): \( \mathbf{a} - \mathbf{b} = 9\mathbf{i} - 3\mathbf{j} - 4\mathbf{k} \)
• Part (c): \( \mathbf{c} = -9\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} \)

Feel free to ask if you have any further questions or need clarification on any of the steps!