If 𝑢⃗ = <10, 2> and 𝑣⃗ =<-3, 4>, then what is the magnitude of  12𝑢⃗ −2𝑣⃗ .

To find the magnitude of the vector \( 12\mathbf{u} - 2\mathbf{v} \), we will follow these steps:

1. **Calculate \( 12\mathbf{u} \) and \( -2\mathbf{v} \)**:
   \[
   12\mathbf{u} = 12 \times \langle 10, 2 \rangle = \langle 120, 24 \rangle
   \]
   \[
   -2\mathbf{v} = -2 \times \langle -3, 4 \rangle = \langle 6, -8 \rangle
   \]

2. **Find the resultant vector \( 12\mathbf{u} - 2\mathbf{v} \)**:
   \[
   12\mathbf{u} - 2\mathbf{v} = \langle 120, 24 \rangle + \langle 6, -8 \rangle = \langle 126, 16 \rangle
   \]

3. **Calculate the magnitude of the resultant vector**:
   The magnitude of a vector \( \mathbf{w} = \langle a, b \rangle \) is given by \( \|\mathbf{w}\| = \sqrt{a^2 + b^2} \).
   \[
   \|\mathbf{w}\| = \sqrt{126^2 + 16^2} = \sqrt{15876 + 256} = \sqrt{16132}
   \]

4. **Simplify the square root**:
   \[
   \sqrt{16132} \approx 126.98
   \]

Thus, the magnitude of \( 12\mathbf{u} - 2\mathbf{v} \) is approximately **126.98**.

Would you like any further details or have any questions?

Here are some related questions:

1. How do you find the direction of a vector?
2. What is the difference between a unit vector and a regular vector?
3. How can you add and subtract vectors graphically?
4. How do you find the dot product of two vectors?
5. What are some applications of vector magnitudes in physics?

**Tip:** When dealing with vector problems, always double-check your arithmetic when scaling and adding vectors.

Learn how to calculate the magnitude of the vector resulting from the operation 12𝑢⃗ − 2𝑣⃗. This problem involves vector operations, including scalar multiplication and vector addition, with a step-by-step solution. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation