Adding and Subtracting Vectors - TutorOne USA

Adding and Subtracting Vectors

Chapter 20

Adding and Subtracting Vectors Concept Explanation:

Vectors can be added or subtracted using component-wise operations. This means you can add or subtract the corresponding components of the vectors.

If you have two vectors:

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

Then:

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

Common Mistakes:

Confusing Order of Operations: Ensure you are correctly adding or subtracting each component.
Forgetting to Maintain Dimensions: Always check if the vectors are of the same dimensions.

Hard Questions:

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

Step-by-Step Solution:

Add the vectors component-wise:

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

Answer:

The result is $\begin{bmatrix} 4 \ 6 \end{bmatrix}$ .

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

Step-by-Step Solution:

Subtract the vectors component-wise:

$\vec{A} - \vec{B} = \begin{bmatrix} 3 - 2 \ 1 - 5 \end{bmatrix} = \begin{bmatrix} 1 \ -4 \end{bmatrix}$

Answer:

The result is $\begin{bmatrix} 1 \ -4 \end{bmatrix}$ .

Q3: Add $\vec{C} = \begin{bmatrix} -3 \ 2 \end{bmatrix}$ and $\vec{D} = \begin{bmatrix} 5 \ -4 \end{bmatrix}$ .

Step-by-Step Solution:

Add the vectors component-wise:

$\vec{C} + \vec{D} = \begin{bmatrix} -3 + 5 \ 2 - 4 \end{bmatrix} = \begin{bmatrix} 2 \ -2 \end{bmatrix}$

Answer:

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

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