Adding and Subtracting Vectors - TutorOne USA

Adding and Subtracting Vectors

Chapter 28

Concept Explanation:

Vectors are quantities that have both magnitude and direction. When adding or subtracting vectors, you perform the operation component-wise.

Given two vectors:

$\vec{A} = \begin{bmatrix} a_1 \ a_2 \end{bmatrix}, \quad \vec{B} = \begin{bmatrix} b_1 \ b_2 \end{bmatrix}$

The addition is:

$\vec{A} + \vec{B} = \begin{bmatrix} a_1 + b_1 \ a_2 + b_2 \end{bmatrix}$

Common Mistakes:

Component Misalignment: Ensure you are adding corresponding components correctly.
Direction Confusion: Remember that subtracting a vector is equivalent to adding its negative.

Helpful Tips:

Graphical Representation: Drawing vectors on a coordinate plane can help visualize the addition or subtraction process.
Use Unit Vectors: Understanding unit vectors can simplify the process of working with vectors.

Hard Questions:

Q1: Add $\vec{A} = \begin{bmatrix} 2 \ 3 \end{bmatrix}$ and $\vec{B} = \begin{bmatrix} 4 \ -1 \end{bmatrix}$ .

Step-by-Step Solution:

Add the vectors component-wise:

$\vec{A} + \vec{B} = \begin{bmatrix} 2 + 4 \ 3 - 1 \end{bmatrix} = \begin{bmatrix} 6 \ 2 \end{bmatrix}$

Answer:

The result is $\begin{bmatrix} 6 \ 2 \end{bmatrix}$ .

Q2: Subtract $\vec{C} = \begin{bmatrix} 5 \ 3 \end{bmatrix}$ from $\vec{D} = \begin{bmatrix} 2 \ 1 \end{bmatrix}$ .

Step-by-Step Solution:

Subtract the vectors component-wise:

$\vec{D} - \vec{C} = \begin{bmatrix} 2 - 5 \ 1 - 3 \end{bmatrix} = \begin{bmatrix} -3 \ -2 \end{bmatrix}$

Answer:

The result is $\begin{bmatrix} -3 \ -2 \end{bmatrix}$ .

Q3: Add $\vec{E} = \begin{bmatrix} 3 \ -2 \end{bmatrix}$ and $\vec{F} = \begin{bmatrix} -1 \ 4 \end{bmatrix}$ .

Step-by-Step Solution:

Add the vectors component-wise:

$\vec{E} + \vec{F} = \begin{bmatrix} 3 - 1 \ -2 + 4 \end{bmatrix} = \begin{bmatrix} 2 \ 2 \end{bmatrix}$

Answer:

The result is $\begin{bmatrix} 2 \ 2 \end{bmatrix}$ .

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