Problem 58 - Precalculus with Calculus Previews

Question

In Problems \(49-62, \mathbf{u}=\langle 2,-3\rangle, \mathbf{v}=\langle-1,5\rangle,\) and \(\mathbf{w}=\langle 3,-2\rangle .\) Find the indicated scalar or vector. $$ (2 \mathbf{v}) \cdot(3 \mathbf{w}) $$

Step-by-Step Solution

Answer

The result of \((2\mathbf{v}) \cdot (3\mathbf{w})\) is \(-78\).

Step 1: Scale the Vector \( \mathbf{v} \)

To scale the vector \( \mathbf{v} = \langle -1, 5 \rangle \) by 2, multiply each component by 2. Thus, \[ 2 \mathbf{v} = 2 \times \langle -1, 5 \rangle = \langle 2 \times -1, 2 \times 5 \rangle = \langle -2, 10 \rangle. \]

Step 2: Scale the Vector \( \mathbf{w} \)

To scale the vector \( \mathbf{w} = \langle 3, -2 \rangle \) by 3, multiply each component by 3. Thus, \[ 3 \mathbf{w} = 3 \times \langle 3, -2 \rangle = \langle 3 \times 3, 3 \times -2 \rangle = \langle 9, -6 \rangle. \]

Step 3: Compute the Dot Product

Use the formula for the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), which is \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2. \]Apply this formula to the vectors \( 2 \mathbf{v} = \langle -2, 10 \rangle \) and \( 3 \mathbf{w} = \langle 9, -6 \rangle \):\[ (2 \mathbf{v}) \cdot (3 \mathbf{w}) = (-2)(9) + (10)(-6) = -18 - 60 = -78. \]

Key Concepts

Vector Scaling

When scaling a vector, you multiply each component of the vector by the same factor. This process changes the vector's magnitude but not its direction. Imagine you have a vector like \( \mathbf{v} = \langle -1, 5 \rangle \) and you're asked to scale it by 2. Here's how you would perform the operation:

• Take the vector \( \mathbf{v} = \langle -1, 5 \rangle \).
• Multiply each component of the vector by 2.
• This results in a new vector: \( 2 \mathbf{v} = \langle 2 \times -1, 2 \times 5 \rangle = \langle -2, 10 \rangle \).

Scaling is a valuable operation because it allows you to easily adjust the size of a vector while keeping its direction the same.
Let's extend this idea to another example, where vector \( \mathbf{w} = \langle 3, -2 \rangle \) gets scaled by 3:

• Start with the vector \( \mathbf{w} = \langle 3, -2 \rangle \).
• Multiply each component of the vector by 3.
• The scaled vector becomes: \( 3 \mathbf{w} = \langle 3 \times 3, 3 \times -2 \rangle = \langle 9, -6 \rangle \).

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors that results in a scalar. To calculate the dot product of vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), use the formula:
\[\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2\]This operation is useful for finding the angle between two vectors or determining if vectors are perpendicular (if their dot product is zero).
Applying the dot product to our scaled vectors:

• We have the vectors \( 2 \mathbf{v} = \langle -2, 10 \rangle \) and \( 3 \mathbf{w} = \langle 9, -6 \rangle \).
• Calculate their dot product: \( (2 \mathbf{v}) \cdot (3 \mathbf{w}) \).
• Using the formula, we compute: \( (-2)(9) + (10)(-6) = -18 - 60 \).
• The result of the dot product is \( -78 \).

Understanding the dot product helps you see how closely aligned two vectors are to each other. In this case, the negative result indicates they point in generally opposite directions.

Vector Arithmetic

Vector arithmetic involves operations like addition, subtraction, and scaling of vectors. These operations allow you to perform calculations directly with vectors, treating them like quantities with both magnitude and direction.

Operations in Vector Arithmetic:

• Addition: To add two vectors, add their corresponding components. For example, if \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), then \( \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2 \rangle \).
• Subtraction: Similar to addition, but you subtract the corresponding components: \( \mathbf{a} - \mathbf{b} = \langle a_1 - b_1, a_2 - b_2 \rangle \).
• Scaling: As previously mentioned, multiplying each component of the vector by a scalar: \( k \mathbf{v} = \langle k v_1, k v_2 \rangle \).

Vector arithmetic is fundamental in physics, engineering, and computer graphics, where calculations with vectors are essential for modeling and analyzing different situations. Remember, understanding these basic operations forms the groundwork for tackling more complex vector problems later.