Problem 67 - 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. $$ \mathbf{v}+2(\mathbf{u}-3 \mathbf{w}) $$

Step-by-Step Solution

Answer

The vector is \( \langle -11, -41, -49 \rangle \).

Step 1: Identify the Expression

We need to find the vector resulting from the expression \( \mathbf{v} + 2(\mathbf{u} - 3 \mathbf{w}) \). Start by identifying clear components: \( \mathbf{v} = \langle -1, 1, 1 \rangle \), \( \mathbf{u} = \langle 1, -3, 2 \rangle \), and \( \mathbf{w} = \langle 2, 6, 9 \rangle \).

Step 2: Calculate \(3\mathbf{w}\)

Multiply each component of \( \mathbf{w} \) with 3: \[ 3 \mathbf{w} = 3 \times \langle 2, 6, 9 \rangle = \langle 6, 18, 27 \rangle \]

Step 3: Compute \(\mathbf{u} - 3\mathbf{w}\)

Subtract the components of \( 3\mathbf{w} \) from \( \mathbf{u} \): \[ \mathbf{u} - 3\mathbf{w} = \langle 1, -3, 2 \rangle - \langle 6, 18, 27 \rangle = \langle 1-6, -3-18, 2-27 \rangle = \langle -5, -21, -25 \rangle \]

Step 4: Compute \(2(\mathbf{u} - 3\mathbf{w})\)

Multiply each component of \( \mathbf{u} - 3\mathbf{w} \) by 2: \[ 2(\mathbf{u} - 3\mathbf{w}) = 2 \times \langle -5, -21, -25 \rangle = \langle -10, -42, -50 \rangle \]

Step 5: Calculate \(\mathbf{v} + 2(\mathbf{u} - 3\mathbf{w})\)

Add the components of \( \mathbf{v} \) and \( 2(\mathbf{u} - 3\mathbf{w}) \): \[ \mathbf{v} + 2(\mathbf{u} - 3\mathbf{w}) = \langle -1, 1, 1 \rangle + \langle -10, -42, -50 \rangle = \langle -1 - 10, 1 - 42, 1 - 50 \rangle = \langle -11, -41, -49 \rangle \]

Key Concepts

Understanding Vector Addition

Vector addition is a process where we combine two vectors to form a new vector. Essentially, we add the corresponding components of the vectors together. This operation follows the basic rules of addition that we use in arithmetic.
For example, when adding two vectors

• Vector A = \( \langle a_1, a_2, a_3 \rangle \)
• Vector B = \( \langle b_1, b_2, b_3 \rangle \)

,
the resulting vector, say Vector C, is obtained by adding the same position elements of Vector A and B.
Mathematically, this can be expressed as:
\[ \text{Vector C} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle \]This operation is useful because it helps combine different effects or movements that are represented as vectors, leading to a resulting direction and magnitude. Always make sure the vectors are in the same dimension when performing this operation.

Exploring Vector Subtraction

Vector subtraction involves taking one vector's components and subtracting them from another vector's. This operation is pretty similar to vector addition.
The only difference is that we are subtracting the components instead of adding them. Suppose we have two vectors:

• Vector D = \( \langle d_1, d_2, d_3 \rangle \)
• Vector E = \( \langle e_1, e_2, e_3 \rangle \)

When Vector D is subtracted by Vector E, the resulting vector, let's call it Vector F, is
\[\text{Vector F} = \langle d_1 - e_1, d_2 - e_2, d_3 - e_3 \rangle\]This process is similar to calculating the difference between two directions or forces.
It is useful in scenarios where we need to calculate resultants or outcomes based on differing vector properties. Remember, subtraction affects both the direction and magnitude of the resultant vector.

Understanding Scalar Multiplication of Vectors

Scalar multiplication means multiplying a vector by a scalar, which is just a fancy term for a real number. The idea is to stretch or shrink a vector by multiplying each of its components by the scalar. Suppose we have a vector \( \mathbf{G} = \langle g_1, g_2, g_3 \rangle \) and a scalar \( s \).
The result of multiplying the vector by the scalar is:
\[\text{Scalar multiplied vector} = s \times \langle g_1, g_2, g_3 \rangle = \langle s \cdot g_1, s \cdot g_2, s \cdot g_3 \rangle\]This operation scales the vector by the amount \( s \). If \( s > 1 \), the vector lengthens, whereas if \( 0 < s < 1 \), it shortens.
If \( s = -1 \), it reverses the direction of the vector. Scalar multiplication has diverse applications, such as scaling forces, directions, or any vector quantities in physics and engineering.