u = 5i + 7j v=-2i - 11j Find: -5u + 2v

Question

Complete the following vector operations.

## Problem Statement:

Given two vectors \( \mathbf{u} \) and \( \mathbf{v} \):

- \(\mathbf{u} = 5\mathbf{i} + 7\mathbf{j}\)
- \(\mathbf{v} = -2\mathbf{i} - 11\mathbf{j}\)

### Objective:
Find the vector \(-5\mathbf{u} + 2\mathbf{v}\).

### Steps:
1. **Calculate \(-5\mathbf{u}\):**
   \[
   -5\mathbf{u} = -5(5\mathbf{i} + 7\mathbf{j}) = -25\mathbf{i} - 35\mathbf{j}
   \]

2. **Calculate \(2\mathbf{v}\):**
   \[
   2\mathbf{v} = 2(-2\mathbf{i} - 11\mathbf{j}) = -4\mathbf{i} - 22\mathbf{j}
   \]

3. **Add the resulting vectors \(-25\mathbf{i} - 35\mathbf{j}\) and \(-4\mathbf{i} - 22\mathbf{j}\):**
   \[
   -5\mathbf{u} + 2\mathbf{v} = (-25\mathbf{i} - 35\mathbf{j}) + (-4\mathbf{i} - 22\mathbf{j}) = -29\mathbf{i} - 57\mathbf{j}
   \]

Thus, \(-5\mathbf{u} + 2\mathbf{v} = -29\mathbf{i} - 57\mathbf{j}\).
Transcribed Image Text:## Problem Statement: Given two vectors \( \mathbf{u} \) and \( \mathbf{v} \): - \(\mathbf{u} = 5\mathbf{i} + 7\mathbf{j}\) - \(\mathbf{v} = -2\mathbf{i} - 11\mathbf{j}\) ### Objective: Find the vector \(-5\mathbf{u} + 2\mathbf{v}\). ### Steps: 1. **Calculate \(-5\mathbf{u}\):** \[ -5\mathbf{u} = -5(5\mathbf{i} + 7\mathbf{j}) = -25\mathbf{i} - 35\mathbf{j} \] 2. **Calculate \(2\mathbf{v}\):** \[ 2\mathbf{v} = 2(-2\mathbf{i} - 11\mathbf{j}) = -4\mathbf{i} - 22\mathbf{j} \] 3. **Add the resulting vectors \(-25\mathbf{i} - 35\mathbf{j}\) and \(-4\mathbf{i} - 22\mathbf{j}\):** \[ -5\mathbf{u} + 2\mathbf{v} = (-25\mathbf{i} - 35\mathbf{j}) + (-4\mathbf{i} - 22\mathbf{j}) = -29\mathbf{i} - 57\mathbf{j} \] Thus, \(-5\mathbf{u} + 2\mathbf{v} = -29\mathbf{i} - 57\mathbf{j}\).
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