Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-28: Given that \(a=\langle 2,-3\rangle, \quad b=\langle 3,4\rangle\), and \(c=\langle-1,5\rangle\), find the number. $$ \operatorname{comp}_{\mathrm{b}}(\mathbf{a}+\mathbf{c}) $$

Step-by-Step Solution

Answer

\( \operatorname{comp}_{\mathrm{b}}(\mathbf{a} + \mathbf{c}) = \frac{11}{5} \)."

Step 1: Find the vector sum \( \mathbf{a} + \mathbf{c} \)

First, we need to find the vector sum of \( \mathbf{a} \) and \( \mathbf{c} \). Given \( a = \langle 2, -3 \rangle \) and \( c = \langle -1, 5 \rangle \), we add these two vectors component-wise: \( \mathbf{a} + \mathbf{c} = \langle 2 + (-1), -3 + 5 \rangle = \langle 1, 2 \rangle \).

Step 2: Calculate the dot product \( (\mathbf{a} + \mathbf{c}) \cdot \mathbf{b} \)

Calculate the dot product of \( \mathbf{a} + \mathbf{c} \) with \( \mathbf{b} \). Using \( \mathbf{b} = \langle 3,4 \rangle \), the dot product is: \[ (\mathbf{a} + \mathbf{c}) \cdot \mathbf{b} = \langle 1, 2 \rangle \cdot \langle 3, 4 \rangle = 1 \cdot 3 + 2 \cdot 4 = 3 + 8 = 11 \].

Step 3: Find the magnitude of \( \mathbf{b} \)

The magnitude of a vector \( \mathbf{b} = \langle 3, 4 \rangle \) is calculated using the formula \( \| \mathbf{b} \| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

Step 4: Compute the scalar component \( \operatorname{comp}_{\mathrm{b}}(\mathbf{a} + \mathbf{c}) \)

Now, use the formula for the scalar component to find \( \operatorname{comp}_{\mathrm{b}}(\mathbf{a} + \mathbf{c}) = \frac{(\mathbf{a} + \mathbf{c}) \cdot \mathbf{b}}{\|\mathbf{b}\|} = \frac{11}{5} \).

Key Concepts

Vector Addition

Vector addition is a fundamental operation in vector algebra that combines two or more vectors to form a new vector. To add vectors, you simply add their corresponding components. For instance, consider vectors \( \mathbf{a}=\langle 2,-3\rangle \) and \( \mathbf{c}=\langle -1,5 \rangle \). You add each of their components separately:\

• The first components: \( 2 + (-1) = 1 \)
• The second components: \( -3 + 5 = 2 \)

Thus, the resulting vector from adding \( \mathbf{a} \) and \( \mathbf{c} \) is \( \langle 1, 2 \rangle \). This operation is quite useful in various applications, where combining different directional quantities is necessary.
Remember that the components of vectors can represent anything from speed and direction to forces, making vector addition a versatile tool in physics and engineering.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a scalar value. It is crucial for determining the angle between vectors and projection measures. To compute the dot product of two vectors, you multiply their corresponding components and sum the results.
For example, using vectors\( (\mathbf{a} + \mathbf{c})=\langle 1,2\rangle \) and \( \mathbf{b}=\langle 3,4\rangle \), the dot product is calculated as follows:\

• Multiply the first components: \( 1 \times 3 = 3 \)
• Multiply the second components: \( 2 \times 4 = 8 \)
• Add the products: \( 3 + 8 = 11 \)

So, the dot product \((\mathbf{a} + \mathbf{c}) \cdot \mathbf{b} = 11\). This product is essential for various calculations, including finding the angles between vectors and calculating projections.

Vector Magnitude

Vector magnitude, or length, is a measure of how long a vector is. It is represented as \( \|\mathbf{b}\| \). To find the magnitude of a vector, you use the Pythagorean theorem in a multi-dimensional space.
For vector \( \mathbf{b} = \langle 3, 4 \rangle \), the magnitude is calculated as follows:\[\|\mathbf{b}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
The concept of magnitude is essential because it allows us to quantify the size of vectors, whether they represent speed, force, or any other vector quantity. A vector's magnitude gives an intuitive sense of its actual size regardless of its direction.

Scalar Projection

Scalar projection is the measure of one vector onto another, quantifying how much one vector extends along the direction of another vector. It is often useful in physics and engineering to determine effective components of forces, velocities, etc. The scalar projection of vector \( \mathbf{a} + \mathbf{c} \) onto \( \mathbf{b} \) is given by the formula: \[ \operatorname{comp}_{\mathbf{b}}(\mathbf{a} + \mathbf{c}) = \frac{(\mathbf{a} + \mathbf{c}) \cdot \mathbf{b}}{\|\mathbf{b}\|} \]
With the dot product already calculated as \( 11 \), and the magnitude of \( \mathbf{b} \) as \( 5 \), it becomes:\

• \( \operatorname{comp}_{\mathbf{b}}(\mathbf{a} + \mathbf{c}) = \frac{11}{5} \)

This computation shows how much of the vector \( \mathbf{a} + \mathbf{c} \) actually projects onto \( \mathbf{b} \), giving a scalar value. Scalar projection helps in understanding the component of a vector in the direction of another.