Problem 75 - Precalculus with Calculus Previews

Question

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ (\mathbf{u}+\mathbf{v}) \cdot \mathbf{w} $$

Step-by-Step Solution

Answer

The result of \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{w}\) is 15.

Step 1: Add the Vectors \( \mathbf{u} \) and \( \mathbf{v} \)

Calculate the vector sum \( \mathbf{u} + \mathbf{v} \) by adding the corresponding components of \( \mathbf{u} = \langle 1, -3, 2 \rangle \) and \( \mathbf{v} = \langle -1, 1, 1 \rangle \). This gives you \( \mathbf{u} + \mathbf{v} = \langle 1 + (-1), -3 + 1, 2 + 1 \rangle = \langle 0, -2, 3 \rangle \).

Step 2: Compute the Dot Product \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{w}\)

To find the dot product, take \( (\mathbf{u} + \mathbf{v}) = \langle 0, -2, 3 \rangle \) and \( \mathbf{w} = \langle 2, 6, 9 \rangle \). The dot product is calculated as \( 0 \times 2 + (-2) \times 6 + 3 \times 9 \). This simplifies to \( 0 - 12 + 27 = 15 \).

Key Concepts

Vector Addition

Vector addition is like adding things to your cart in a store. Each item is a vector, and you're adding them up one by one. When you add two vectors, like \( \mathbf{u} = \langle 1, -3, 2 \rangle \) and \( \mathbf{v} = \langle -1, 1, 1 \rangle \), you add the first components together, then the second components, and finally the third components. This is similar to adding items from different aisles but eventually putting them into the same cart. Here, adding \( \mathbf{u} \) and \( \mathbf{v} \) gives us \( \mathbf{u} + \mathbf{v} = \langle 1 + (-1), -3 + 1, 2 + 1 \rangle \).
After doing the math, you get \( \langle 0, -2, 3 \rangle \). Now, this is like having a summary of everything you added together. Vector addition is straightforward once you understand that you are just combining components. It respects the rule: sum them one by one, component by component.

• Component 1: Add \(1 + (-1) = 0\).
• Component 2: Add \(-3 + 1 = -2\).
• Component 3: Add \(2 + 1 = 3\).

And that’s how you perform vector addition!

Vectors

Vectors are like arrows that point in a direction and have a certain length. Think of them as a way to move from one point to another in space. When we talk about vectors, we often use the form \( \langle x, y, z \rangle \), representing movement in a three-dimensional space.

• They have direction: pointing from one place to another.
• They have magnitude: the length of the arrow gives you an idea of how far you've gone.

For example, the vector \( \mathbf{u} = \langle 1, -3, 2 \rangle \) means you could move 1 unit in the x-direction, -3 units (or backwards) in the y-direction, and 2 units in the z-direction.
Vectors are essential in physics and engineering because they precisely describe motion and forces. Whenever you're performing vector operations, remember that you're dealing with these directed arrows. Each vector tells a story of direction and distance.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a single number, called a scalar. This is like scaling your vector up or down. Imagine you have a vector \( \mathbf{v} = \langle 2, 6, 9 \rangle \), and you need to multiply it by 3. Each component of the vector is multiplied by the scalar 3.
This operation changes the size of the vector but not its direction, like zooming in or out on a picture. The process is simple:

• Multiply each component of the vector by the scalar.
• Result: \( 3 \cdot \mathbf{v} = 3 \cdot \langle 2, 6, 9 \rangle = \langle 6, 18, 27 \rangle \).

Notice how the magnitude of the vector changes. If the scalar is positive, the direction remains the same; if it's negative, the direction reverses. Scalar multiplication allows us to adjust the vector's size, making it an essential tool in scaling forces and other physical quantities.