Problem 23 - 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. $$ (2 \mathbf{a}+\mathbf{b}) \cdot(3 \mathbf{c}) \quad 24(\mathbf{a}-\mathbf{b}) \cdot(\mathbf{b}+\mathbf{c}) $$

Step-by-Step Solution

Answer

The result of the expression is -1611.

Step 1: Understand the Operation Inside Parentheses

First, we need to find the vector operations inside each parentheses. Start with \( 2\mathbf{a} + \mathbf{b} \). Multiply vector \( \mathbf{a} = \langle 2, -3 \rangle \) by 2: \( 2\mathbf{a} = \langle 4, -6 \rangle \). Now add vector \( \mathbf{b} = \langle 3, 4 \rangle \) to it: \[ 2\mathbf{a} + \mathbf{b} = \langle 4 + 3, -6 + 4 \rangle = \langle 7, -2 \rangle \].

Step 2: Calculate the Scalar Multiplication

Next, calculate \( 3\mathbf{c} \). You multiply vector \( \mathbf{c} = \langle -1, 5 \rangle \) by 3: \[ 3\mathbf{c} = \langle -3, 15 \rangle \].

Step 3: Calculate the Dot Product of Two Vectors

Now find the dot product \((2\mathbf{a} + \mathbf{b}) \cdot (3\mathbf{c})\). Use: \( \langle a_1, a_2 \rangle \cdot \langle b_1, b_2 \rangle = a_1b_1 + a_2b_2 \). For \( \langle 7, -2 \rangle \cdot \langle -3, 15 \rangle \), calculate: \[ 7(-3) + (-2)(15) = -21 - 30 = -51 \].

Step 4: Calculate vector subtraction

Consider \( \mathbf{a} - \mathbf{b} \). You subtract vector \( \mathbf{b} = \langle 3, 4 \rangle \) from vector \( \mathbf{a} = \langle 2, -3 \rangle \): \( \mathbf{a} - \mathbf{b} = \langle 2 - 3, -3 - 4 \rangle = \langle -1, -7 \rangle \).

Step 5: Calculate vector addition

Next, calculate \( \mathbf{b} + \mathbf{c} \). Add vector \( \mathbf{b} = \langle 3, 4 \rangle \) and vector \( \mathbf{c} = \langle -1, 5 \rangle \): \( \mathbf{b} + \mathbf{c} = \langle 3 + (-1), 4 + 5 \rangle = \langle 2, 9 \rangle \).

Step 6: Calculate the Second Dot Product

Now find \(( \mathbf{a} - \mathbf{b}) \cdot (\mathbf{b} + \mathbf{c})\). Use the same dot product formula as in Step 3: \( \langle -1, -7 \rangle \cdot \langle 2, 9 \rangle = (-1)(2) + (-7)(9) = -2 - 63 = -65 \).

Step 7: Multiply by 24

Finally, multiply the dot product by 24: \(24 \times (-65) = -1560\).

Step 8: Calculate the Final Expression

Combine your results from Step 3 and Step 7: \(-51 - 1560 = -1611\). Thus, the value of the expression is \(-1611\).

Key Concepts

Dot Product

The dot product is a fascinating operation in vector algebra that results in a scalar, rather than a vector. Given two vectors, say \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), their dot product is calculated as \( u_1 \times v_1 + u_2 \times v_2 \). It's useful because it allows us to determine the angle between vectors or evaluate projections in physics.
For example, in our exercise, we calculated \((2\mathbf{a} + \mathbf{b}) \cdot (3\mathbf{c})\) and found it to be \(-51\). Breaking it down step-by-step:

• First, find the vectors involved.
• Compute their components' products.
• Sum these products to get a single number (a scalar).

That's the beauty of the dot product—it boils down often complex vector interactions into single, easy-to-work-with numbers.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (or real number) and stretches or shrinks the vector's magnitude while preserving its direction. This is handy when you need to adjust a vector's length proportionally.
Consider vector \( \mathbf{c} = \langle -1, 5 \rangle \) from our exercise. Multiplying this by the scalar 3, we get \( 3\mathbf{c} = \langle -3, 15 \rangle \).

• The components are multiplied individually by the scalar: the first component, \(-1\), becomes \(-3\), and the second component, 5, becomes 15.

This process changes the vector's size while its direction remains unchanged. Scalar multiplication is fundamental when scaling vectors in geometry and physics.

Vector Addition

Vector addition is both intuitive and straightforward. It involves adding corresponding components of two or more vectors together. This operation results in a new vector that represents the cumulative effect of adding the initial vectors.
In our steps, adding \( \mathbf{b} = \langle 3, 4 \rangle \) with \( \mathbf{c} = \langle -1, 5 \rangle \) yields a new vector \( \mathbf{b} + \mathbf{c} = \langle 2, 9 \rangle \).

• Add the respective components: \( (3 + (-1)) = 2 \) and \( (4 + 5) = 9 \).

Notice how each component of the result is the sum of corresponding components from the vectors being added. This simple operation is crucial in many fields like engineering and computer graphics for modeling motion or forces.

Vector Subtraction

Vector subtraction, much like vector addition, involves operating on the individual components of vectors to produce another vector. Here, you subtract the respective components of one vector from another. It's useful for tasks such as finding the difference in direction or calculating displacements.
For example, in our solution, vector \( \mathbf{a} = \langle 2, -3 \rangle \) is subtracted by vector \( \mathbf{b} = \langle 3, 4 \rangle \), resulting in \( \mathbf{a} - \mathbf{b} = \langle -1, -7 \rangle \).

• Subtract each component: \( 2 - 3 = -1 \) and \( -3 - 4 = -7 \).

Through this operation, one can easily determine the resultant vector that indicates how one initial vector has been transformed relative to another. Vector subtraction is widely used in physics for determining forces or velocities.