Question

Let \(A=2 \mathrm{i}+3 \mathrm{j}\) and \(\mathrm{B}=4 \mathrm{i}-5 \mathrm{j} .\) Show graphically, and find algebraically, the vectors \(-\mathrm{A}\), \(3 \mathbf{B}, \mathbf{A}-\mathbf{B}, \mathbf{B}+2 \mathbf{A}, \frac{1}{2}(\mathbf{A}+\mathbf{B})\)

Short answer

\(-A = -2i - 3j\), \(3B = 12i - 15j\), \(A - B = -2i + 8j\), \(B + 2A = 8i + j\), \(\frac{1}{2}(A + B) = 3i - j\)

Step-by-step solution

Understanding the Vectors A and B

Given vectors are \(A = 2i + 3j\) and \(B = 4i - 5j\). These vectors need to be manipulated algebraically.

Finding -A

To find \(-A\), multiply each component of \(A\) by -1. \(-A = -1 \cdot (2i + 3j) = -2i - 3j\)

Finding 3B

To find \(3B\), multiply each component of \(B\) by 3. \(3B = 3 \cdot (4i - 5j) = 12i - 15j\)

Finding A - B

To find \(A - B\), subtract the components of \(B\) from the components of \(A\). \(A - B = (2i + 3j) - (4i - 5j) = (2i - 4i) + (3j + 5j) = -2i + 8j\)

Finding B + 2A

To find \(B + 2A\), first calculate \(2A\) and then add it to \(B\). \(2A = 2 \cdot (2i + 3j) = 4i + 6j\), therefore \(B + 2A = (4i - 5j) + (4i + 6j) = (4i + 4i) + (-5j + 6j) = 8i + j\)

Finding \(\frac{1}{2}(A + B)\)

To find \(\frac{1}{2}(A + B)\), first calculate \(A + B\) and then divide by 2. \(A + B = (2i + 3j) + (4i - 5j) = (2i + 4i) + (3j - 5j) = 6i - 2j\), therefore \(\frac{1}{2}(A + B) = \frac{1}{2} (6i - 2j) = 3i - j\)

Key concepts

vector addition

Vector addition is an essential concept in both physics and mathematics. When adding two vectors, you combine their corresponding components. For instance, if we have vector \(A = 2i + 3j\) and vector \(B = 4i - 5j\), to find \(A + B\), simply add the i-components together and the j-components together.
So, \(A + B = (2i + 3j) + (4i - 5j) = (2i + 4i) + (3j - 5j) = 6i - 2j\).
This means that when |\text|vectors A and B|vector| interact together, we get a new vector represented by \(6i - 2j\). This new vector describes a combined effect of both A and B.

scalar multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a single number). For instance, for vector \(B = 4i - 5j\), if we multiply B by 3, every component of the vector is scaled by the number 3.
This gives \(3B = 3 \times (4i - 5j) = 12i - 15j\).
Scalar multiplication stretches or compresses a vector depending on the value of the scalar. Positive scalars keep the direction unchanged, while negative scalars reverse it.

vector subtraction

Vector subtraction is similar to vector addition but involves subtracting the components of one vector from the components of another. To find \(A - B\) where \(A = 2i + 3j\) and \(B = 4i - 5j\), subtract the i-components and j-components separately.
So, \(A - B = (2i + 3j) - (4i - 5j) = (2i - 4i) + (3j + 5j) = -2i + 8j\).
This results in a vector that represents the difference between vectors A and B.

graphical representation

Graphically representing vectors helps in visual comprehension. Each vector can be depicted as an arrow with a starting point (origin) and an endpoint. The length of the arrow indicates the vector's magnitude, while the direction shows its orientation.
For instance, vector \(A = 2i + 3j\) is drawn from the origin to the point (2,3). Similarly, for vector \(B = 4i - 5j\), it starts at the origin and ends at (4, -5).
When vectors are manipulated (like added or subtracted), their graphical representations also shift accordingly, showcasing the new resultant vector.

physics mathematics

Vectors are crucial in physics mathematics, especially in describing quantities having both magnitude and direction. Examples include displacement, velocity, force, and acceleration.
For problem-solving, understanding vector manipulation—like addition, subtraction, and scalar multiplication—is vital.
In the given exercise, we used these operations to find new vectors from the given ones. Such skills are foundational in physics to analyze and predict physical phenomena accurately.