Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Find, if possible, \(A+B, A-B, 2 A\), and \(-3 B\) $$ A=\left[\begin{array}{r} 7 \\ -16 \end{array}\right], \quad B=\left[\begin{array}{r} -11 \\ 9 \end{array}\right] $$

Step-by-Step Solution

Answer

\(A+B = \begin{bmatrix} -4 \\ -7 \end{bmatrix}\), \(A-B = \begin{bmatrix} 18 \\ -25 \end{bmatrix}\), \(2A = \begin{bmatrix} 14 \\ -32 \end{bmatrix}\), \(-3B = \begin{bmatrix} 33 \\ -27 \end{bmatrix}\).

Step 1: Addition of Vectors

Before performing vector addition, ensure that both vectors have the same dimension. The vectors \(A\) and \(B\) are both 2-dimensional. Add the corresponding components: \[A + B = \begin{bmatrix} 7 \ -16 \end{bmatrix} + \begin{bmatrix} -11 \ 9 \end{bmatrix} = \begin{bmatrix} 7 + (-11) \ -16 + 9 \end{bmatrix} = \begin{bmatrix} -4 \ -7 \end{bmatrix} \]

Step 2: Subtraction of Vectors

Subtract vector \(B\) from vector \(A\) by subtracting each corresponding component:\[A - B = \begin{bmatrix} 7 \ -16 \end{bmatrix} - \begin{bmatrix} -11 \ 9 \end{bmatrix} = \begin{bmatrix} 7 - (-11) \ -16 - 9 \end{bmatrix} = \begin{bmatrix} 18 \ -25 \end{bmatrix} \]

Step 3: Scalar Multiplication of \(A\)

Multiply each component of vector \(A\) by the scalar 2:\[2A = 2 \cdot \begin{bmatrix} 7 \ -16 \end{bmatrix} = \begin{bmatrix} 2 \cdot 7 \ 2 \cdot (-16) \end{bmatrix} = \begin{bmatrix} 14 \ -32 \end{bmatrix} \]

Step 4: Scalar Multiplication of \(B\)

Multiply each component of vector \(B\) by the scalar -3:\[-3B = -3 \cdot \begin{bmatrix} -11 \ 9 \end{bmatrix} = \begin{bmatrix} -3 \cdot (-11) \ -3 \cdot 9 \end{bmatrix} = \begin{bmatrix} 33 \ -27 \end{bmatrix} \]

Key Concepts

Vector Addition

Vector addition is one of the fundamental operations in linear algebra. It involves the combination of two vectors, resulting in a new vector. This operation can be visualized as placing the tail of the second vector at the head of the first vector and then drawing a vector from the tail of the first to the head of the second.

• Make sure both vectors have the same dimension before adding. In this context, vector addition requires that each component of one vector is matched with the corresponding component of the other vector.
• For vectors \( A = [a_1, a_2] \) and \( B = [b_1, b_2] \), the sum \( A + B \) is \([a_1 + b_1, a_2 + b_2]\).

Here is an example:
If \( A = \begin{bmatrix} 7 \ -16 \end{bmatrix} \) and \( B = \begin{bmatrix} -11 \ 9 \end{bmatrix} \), then:

• Add 7 and -11 to get -4.
• Add -16 and 9 to get -7.
• The result is the vector: \( \begin{bmatrix} -4 \ -7 \end{bmatrix} \).

Vector Subtraction

Similar to addition, vector subtraction deals with vectors of the same dimensions, and it is essentially the addition of a negative vector. Subtraction can be imagined as adding the negative of a vector, reflecting it across the origin before performing addition.

• For vectors \( A = [a_1, a_2] \) and \( B = [b_1, b_2] \), the difference \( A - B \) is \([a_1 - b_1, a_2 - b_2]\).

Let's use the vectors from before:
Given \( A = \begin{bmatrix} 7 \ -16 \end{bmatrix} \) and \( B = \begin{bmatrix} -11 \ 9 \end{bmatrix} \), the subtraction is calculated as:

• Subtract -11 from 7 to get 18.
• Subtract 9 from -16 to get -25.
• The resultant vector is \( \begin{bmatrix} 18 \ -25 \end{bmatrix} \).

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a single number, or scalar. This operation stretches or shrinks the vector. If the scalar is negative, the vector's direction is reversed.

• For a vector \( A = [a_1, a_2] \) and scalar \( c \), the product \( cA \) is \([c \, a_1, c \, a_2]\).

Consider these examples:
For vector \( A = \begin{bmatrix} 7 \ -16 \end{bmatrix} \) and scalar 2, the result:

• Multiply each component by 2.
• The result is \( \begin{bmatrix} 14 \ -32 \end{bmatrix} \).

For scalar -3 applied to vector \( B = \begin{bmatrix} -11 \ 9 \end{bmatrix} \):

• Multiply each component by -3.
• This yields \( \begin{bmatrix} 33 \ -27 \end{bmatrix} \).

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors and operations on them, which includes vector addition, subtraction, and scalar multiplication. It underpins many areas of science and engineering, providing tools for modeling both simple and complex systems.

• Vectors are like arrows that have magnitude and direction, crucial in various scientific computations.
• Understanding these operations allows for the solving of complex problems like those found in physics, computer graphics, and machine learning.

In everyday applications:

• Linear algebra helps model phenomena with equations and matrices.
• It is fundamental in algorithms that solve systems of linear equations and transform geometric shapes.

Mastery of vector operations forms the backbone of linear algebra, leading to advanced topics like matrix transformations, eigenvectors, and orthogonality.