Problem 44 - Algebra and Trigonometry

Question

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(3 \mathbf{v}-2 \mathbf{w}\)

Step-by-Step Solution

Answer

(5, -9)

Step 1: Express \mathbf{v} and \mathbf{w} Separately

Given \( \mathbf{v} = 3 \mathbf{i} - 5 \mathbf{j} \) and \( \mathbf{w} = -2 \mathbf{i} + 3 \mathbf{j} \), write these vectors in component form:\( \mathbf{v} = (3, -5) \)\( \mathbf{w} = (-2, 3) \)

Step 2: Scale Each Vector

Calculate \( 3 \mathbf{v} \) by multiplying \( \mathbf{v} \) by 3:\( 3 \mathbf{v} = 3(3, -5) = (9, -15) \)Calculate \( -2 \mathbf{w} \) by multiplying \( \mathbf{w} \) by -2:\( -2 \mathbf{w} = -2(-2, 3) = (4, -6) \)

Step 3: Subtract the Scaled Vectors

Combine the results from Step 2 into the expression for subtraction:\( 3 \mathbf{v} - 2 \mathbf{w} = (9, -15) - (4, -6) \)Subtract the corresponding components:\[ (9 - 4, -15 - (-6)) \]Simplify the subtraction:\[ (5, -9) \]

Key Concepts

Vector Addition

Vectors are mathematical objects that have both magnitude and direction. They can be added together to produce a new vector. In this problem, we're not directly adding the vectors but we'll still follow similar steps.

When adding two vectors, you simply add their corresponding components. If you have vectors \(\textbf{v} = (v_1, v_2) \) and \(\textbf{w} = (w_1, w_2)\), their sum will be \(\textbf{v} + \textbf{w} = (v_1 + w_1, v_2 + w_2)\).

For example, if \(\textbf{v} = (3, -5)\) and \(\textbf{w} = (-2, 3)\), then:

\( \textbf{v} + \textbf{w} = (3 + (-2), -5 + 3) = (1, -2)\)

Vector addition is essential in many fields such as physics, engineering, and computer science. Understanding it deeply helps simplify complex problems involving multiple directions and magnitudes.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single number). This operation enlarges or shrinks the vector without changing its direction (except by reversing it in case of a negative scalar).

For a vector \(\textbf{v} = (v_1, v_2)\) and a scalar \( c \), the product \( c \textbf{v} \) is calculated as \( c \textbf{v} = (c * v_1, c * v_2)\)

In our problem, we need to calculate \(3 \textbf{v}\) and \(-2 \textbf{w}\):

• \( 3 \textbf{v} = 3 * (3, -5) = (9, -15)\)
• \( -2 \textbf{w} = -2 * (-2, 3) = (4, -6)\)

Scalar multiplication enables easy manipulation of vectors in various scenarios, like determining forces or scaling coordinates in graphics.

Component Form

The component form of a vector is a way to express the vector as an ordered pair or tuple of its individual coordinates. Each component represents how far the vector goes in each dimension.

If a vector is represented as \( \textbf{v} = (v_1, v_2)\), the two values \( v_1 \) and \( v_2 \) are its components in the x and y directions, respectively.

Given\(\textbf{v} = 3\textbf{i} - 5\textbf{j}\) and \(\textbf{w} = -2\textbf{i} + 3\textbf{j} \), we can write them in component form as:

• \( \textbf{v} = (3, -5) \)
• \( \textbf{w} = (-2, 3) \)

Component form makes it easier to conduct further operations like addition and scalar multiplication by dealing with numerical values directly.

Considering our problem, the final vector \(3 \textbf{v} - 2 \textbf{w} \) is simplified as:

\( (9, -15) - (4, -6) = (5, -9)\).