Problem 59 - 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. $$ \mathbf{u} \cdot(\mathbf{u}+\mathbf{v}+\mathbf{w}) $$

Step-by-Step Solution

Answer

The scalar product is 8.

Step 1: Understand the Dot Product

The dot product between two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is calculated using the formula \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \). This will be used to find \( \mathbf{u} \cdot (\mathbf{u} + \mathbf{v} + \mathbf{w}) \).

Step 2: Find the Result of the Vector Addition

Compute the sum of the vectors \( \mathbf{u}, \mathbf{v}, \text{ and } \mathbf{w} \). First, add their components individually: \( \mathbf{u} + \mathbf{v} + \mathbf{w} = \langle 2, -3 \rangle + \langle -1, 5 \rangle + \langle 3, -2 \rangle \). Start by adding \( \langle 2 + (-1), -3 + 5 \rangle = \langle 1, 2 \rangle \). Then add \( \langle 1, 2 \rangle \) and \( \langle 3, -2 \rangle \) to obtain \( \langle 1+3, 2-2 \rangle = \langle 4, 0 \rangle \).

Step 3: Compute the Dot Product

Now compute \( \mathbf{u} \cdot (\mathbf{u} + \mathbf{v} + \mathbf{w}) \) using \( \mathbf{u} = \langle 2, -3 \rangle \) and the resulting vector from step 2, \( \langle 4, 0 \rangle \). The dot product is \( 2 \times 4 + (-3) \times 0 = 8 + 0 = 8 \).

Step 4: Write the Final Answer

The result of the dot product is \( 8 \). Hence, \( \mathbf{u} \cdot (\mathbf{u} + \mathbf{v} + \mathbf{w}) = 8 \).

Key Concepts

Vector Addition

Vector addition is a fundamental operation in mathematics used to combine two or more vectors into a single vector. Imagine vegetables being tossed into a salad bowl; adding vectors works quite similarly.When you add vectors, you simply sum their corresponding components. For example, if you have vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), their sum is another vector, \( \mathbf{c} = \langle a_1 + b_1, a_2 + b_2 \rangle \).In our exercise, the task was to add \( \mathbf{u} = \langle 2, -3 \rangle \), \( \mathbf{v} = \langle -1, 5 \rangle \), and \( \mathbf{w} = \langle 3, -2 \rangle \). First, add \( \mathbf{u} \) and \( \mathbf{v} \) to get the intermediate result \( \langle 1, 2 \rangle \). Next, add this result to \( \mathbf{w} \), arriving at the final combined vector \( \langle 4, 0 \rangle \).

Properties of Vector Addition

• Commutative property: \( \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \)
• Associative property: \( (\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c}) \)
• Adding the zero vector does not change a vector: \( \mathbf{a} + \mathbf{0} = \mathbf{a} \)

Vector Components

Vector components are simply the individual elements or parts of a vector. Picture a vector as an arrow pointing from one location to another. The components tell you exactly how far and in what direction you move along each axis.For a 2-dimensional vector \( \mathbf{v} = \langle v_1, v_2 \rangle \), \( v_1 \) and \( v_2 \) represent the distance along the horizontal (x-axis) and vertical (y-axis) directions, respectively.

Understanding the Role of Components

The components are crucial for various vector operations:

• They allow you to easily compute the vector's magnitude (or length) using the formula \( \sqrt{v_1^2 + v_2^2} \).
• They make adding vectors straightforward, as each component can simply be added to its counterpart from another vector.
• The components are used in calculating dot products and determining angles between vectors.

In solving our exercise, we used these components to add vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \), breaking down a seemingly complex problem into simpler parts that are easy to manage.

Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). It's like scaling a toy; if you double the size, all parts grow twice as large!If you have a vector \( \mathbf{v} = \langle v_1, v_2 \rangle \) and a scalar \( c \), then after scalar multiplication, you get a new vector \( \mathbf{c} = \langle c \cdot v_1, c \cdot v_2 \rangle \).

Impacts of Scalar Multiplication

• Alters the vector's magnitude, making it larger or smaller depending on the scalar's value.
• May change the vector's direction if the scalar is negative, flipping it around its origin.
• Retains the vector’s direction if the scalar is positive.

In our original problem, we used the idea of multiplier effects, though not directly needed in finding the dot product. Scalar multiplication influences practices like vector moving and stretching, vital concepts in physics and engineering situations.