Problem 54 - 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. $$ \mathbf{u} \cdot(\mathbf{v}+\mathbf{w}) $$

Step-by-Step Solution

Answer

The result is \(-5\).

Step 1: Add vectors \(\mathbf{v}\) and \(\mathbf{w}\)

First, compute the sum of the vectors \(\mathbf{v}\) and \(\mathbf{w}\). The vector \(\mathbf{v}\) is \(\langle -1, 5 \rangle\) and \(\mathbf{w}\) is \(\langle 3, -2 \rangle\). Add them component-wise: \(\langle -1 + 3, 5 - 2 \rangle = \langle 2, 3 \rangle\).

Step 2: Calculate the dot product

Now, compute the dot product of \(\mathbf{u}\) with the result of \(\mathbf{v} + \mathbf{w}\). \(\mathbf{u}\) is \(\langle 2, -3 \rangle\) and \(\mathbf{v} + \mathbf{w}\) is \(\langle 2, 3 \rangle\). The dot product is calculated as follows: \((2)(2) + (-3)(3) = 4 - 9 = -5\).

Key Concepts

Vector Addition

When dealing with vectors, one fundamental operation is vector addition. It's a way of combining two vectors to form a new vector. This operation is crucial in understanding more complex vector concepts, such as the dot product.

To add two vectors, say \( \mathbf{v} = \langle a_1, b_1 \rangle \) and \( \mathbf{w} = \langle a_2, b_2 \rangle \),we simply add their corresponding components. This means:

• The first component of the resulting vector is \( a_1 + a_2 \)
• The second component is \( b_1 + b_2 \)

For example, if \( \mathbf{v} = \langle -1, 5 \rangle \)and \( \mathbf{w} = \langle 3, -2 \rangle \), following the rule of component-wise addition,we get\( \langle -1 + 3, 5 - 2 \rangle = \langle 2, 3 \rangle \).

This vector addition sets up many other operations in vector mathematics, ensuring our subsequent operations hold correctly.

Scalar Multiplication

Scalar multiplication is a simple yet powerful vector operation. It involves multiplying each component of a vector by a scalar (a real number). This operation changes the magnitude of the vector but not its direction, unless the scalar is negative, which would reverse it.

If you have a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \)and a scalar \( k \),the scalar multiplication is performed as follows:

• Multiply each component of the vector by the scalar: \( \langle k imes a_1, k imes a_2 \rangle \)

For example, multiplying \( \mathbf{u} = \langle 2, -3 \rangle \)by \( k = 2 \)results in \( \langle 2 imes 2, 2 imes (-3) \rangle = \langle 4, -6 \rangle \).

Scalar multiplication is particularly important when the vector results need to be adjusted by a certain factor or when calculating projections.

Vector Operations

Vector operations encompass a wide range of calculations that can be performed on vectors, including addition, subtraction, scalar multiplication, and the dot product. Understanding these operations is essential for solving problems in physics, engineering, and computer graphics, among other fields.

Dot Product
One key operation is the dot product, which multiplies corresponding components of two vectors and sums the results, giving a scalar. The dot product can determine angles between vectors or project one vector onto another. For vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \)and \( \mathbf{b} = \langle b_1, b_2 \rangle \),the dot product is calculated as follows:

• \( a_1 imes b_1 + a_2 imes b_2 \)

In our exercise, the dot product between \( \mathbf{u} = \langle 2, -3 \rangle \)and \( \mathbf{v} + \mathbf{w} = \langle 2, 3 \rangle \)is calculated as \( 2 imes 2 + (-3) imes 3 = 4 - 9 = -5 \).

Order of Operations
It's crucial to perform vector operations in the correct order, especially when they're combined in problems, like computing the dot product after vector addition. Misordering these steps could result in errors, showing the necessity for a clear understanding of each operation's role. By mastering these operations, you'll establish a strong foundation for tackling complex vector-related challenges.