Question

If \(\|\mathbf{v}\|=4,\) what is the magnitude of \(\frac{1}{2} \mathbf{v}+3 \mathbf{v} ?\)

Short answer

The magnitude is 14.

Step-by-step solution

Express the vector equation

The given vector equation is \ \(\frac{1}{2} \mathbf{v} + 3 \mathbf{v} \). Combine the terms involving \( \mathbf{v} \).

Combine like terms

Add the coefficients of \( \mathbf{v} \) together: \ \( \frac{1}{2} \mathbf{v} + 3 \mathbf{v} = \left( \frac{1}{2} + 3 \right) \mathbf{v} = \left( 3.5 \right) \mathbf{v} \). The simplified expression is \( 3.5 \mathbf{v} \).

Find the magnitude

To find the magnitude of \( 3.5 \mathbf{v} \), use the property that \( \| k \mathbf{v} \| = |k| \cdot \| \mathbf{v} \| \) for any scalar \( k \). Here, \( k = 3.5 \) and \( \| \mathbf{v} \| = 4 \). Hence, \( \| 3.5 \mathbf{v} \| = |3.5| \cdot 4 \).

Calculate the magnitude

Calculate the product: \( |3.5| \times 4 = 14 \). Therefore, the magnitude of \( \frac{1}{2} \mathbf{v} + 3 \mathbf{v} \) is 14.

Key concepts

Vector Operations

Vectors play a crucial role in various fields such as physics, engineering, and computer science.
Vector operations involve adding, subtracting, and multiplying vectors to find resultant vectors or to perform transformations.
Vectors can be represented in different forms like components, magnitude-direction, or unit vectors.
The main operations we deal with are vector addition, scalar multiplication, and finding the magnitude.
Each operation follows specific mathematical rules to simplify or transform the vector expressions.

Scalar Multiplication

Scalar multiplication is when we multiply a vector by a scalar (a real number).
This operation scales the vector either by stretching or compressing it.
The direction of the vector remains unchanged unless the scalar is negative, which would reverse the direction.
The mathematical rule is simple: if we multiply vector \( \mathbf{v} \) by a scalar k, we get \( k \mathbf{v} \).
For example, multiplying a vector by 2 doubles its length, while multiplying by 0.5 halves its length.

Vector Addition

Vector addition is combining two or more vectors to get a single resultant vector.
This operation is straightforward: we add corresponding components of the vectors.
For instance, if we have vectors \( \mathbf{a}=(a_x, a_y) \) and \( \mathbf{b}=(b_x, b_y) \), their sum \( \mathbf{a} + \mathbf{b} \) is \( (a_x + b_x, a_y + b_y) \).
In the problem, we combined \( \frac{1}{2} \mathbf{v} \) and \( 3 \mathbf{v} \) by adding coefficients to get \( 3.5 \mathbf{v} \).
This step simplifies complex vector expressions and is essential for solving vector-related problems.

Magnitude of a Vector

The magnitude of a vector represents its length and is denoted by \( \| \mathbf{v} \| \).
For a vector \( \mathbf{v} = (v_x, v_y) \), the magnitude is calculated as \( \sqrt{v_x^2 + v_y^2} \).
When multiplying by a scalar, the magnitude changes proportionally.
If \( k \) is the scalar and \( \mathbf{v} \) is the original vector, then \( \| k \mathbf{v} \| = |k| \cdot \| \mathbf{v} \| \).
In the example, the magnitude of \( 3.5 \mathbf{v} \) is \( 3.5 \cdot 4 = 14 \).
This formula helps find the scaled magnitude after scalar multiplication.