Problem 66 - 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. $$ 2 \mathbf{u}-(\mathbf{v}-\mathbf{w}) $$

Step-by-Step Solution

Answer

The vector is \( \langle 5, -1, 12 \rangle \).

Step 1: Calculate 2u

Multiply each component of \( \mathbf{u} = \langle 1, -3, 2 \rangle \) by 2. \[ 2 \mathbf{u} = 2 \times \langle 1, -3, 2 \rangle = \langle 2 \times 1, 2 \times -3, 2 \times 2 \rangle = \langle 2, -6, 4 \rangle \]

Step 2: Calculate v - w

Subtract vector \( \mathbf{w} = \langle 2, 6, 9 \rangle \) from vector \( \mathbf{v} = \langle -1, 1, 1 \rangle \) component-wise.\[ \mathbf{v} - \mathbf{w} = \langle -1 - 2, 1 - 6, 1 - 9 \rangle = \langle -3, -5, -8 \rangle \]

Step 3: Calculate 2u - (v - w)

Subtract the result from Step 2 from the result of Step 1 component-wise.\[ 2\mathbf{u} - (\mathbf{v} - \mathbf{w}) = \langle 2, -6, 4 \rangle - \langle -3, -5, -8 \rangle = \langle 2 + 3, -6 + 5, 4 + 8 \rangle = \langle 5, -1, 12 \rangle \]

Key Concepts

Vector Subtraction

Vector subtraction is a straightforward process that involves subtracting the corresponding components of two vectors. Imagine two vectors, \( \mathbf{v} = \langle -1, 1, 1 \rangle \) and \( \mathbf{w} = \langle 2, 6, 9 \rangle \). To find \( \mathbf{v} - \mathbf{w} \), you subtract each component of \( \mathbf{w} \) from the corresponding component of \( \mathbf{v} \):

• First component: \( -1 - 2 = -3 \)
• Second component: \( 1 - 6 = -5 \)
• Third component: \( 1 - 9 = -8 \)

Thus, the resultant vector from this subtraction becomes \( \langle -3, -5, -8 \rangle \). This method helps in understanding how different directions and magnitudes between vectors affect subtraction, making it a crucial skill in vector calculus.
Vector subtraction can often be visualized graphically, where aligning the tail of the second vector to the head of the first makes it easier to grasp the resultant vector conceptually.

Scalar Multiplication

Scalar multiplication is an operation that alters the size of a vector, but not its direction. It's like resizing a vector while maintaining its orientation. In our original exercise, multiplying vector \( \mathbf{u} = \langle 1, -3, 2 \rangle \) by a scalar, which in this case is 2, involves multiplying each component of the vector by the scalar:

• First component: \( 2 \times 1 = 2 \)
• Second component: \( 2 \times -3 = -6 \)
• Third component: \( 2 \times 2 = 4 \)

This results in the vector \( \langle 2, -6, 4 \rangle \).
A useful way to think about scalar multiplication is through the idea of scaling - the vector becomes longer or shorter depending on the scalar, without changing its direction. Scalar multiplication is integral in physics and engineering when you need to scale forces or velocities. It forms the basis of understanding vector spaces, where not only individual vectors but entire spaces can be scaled through scalar multiplication.

Vector Addition

Vector addition combines two or more vectors to yield a resultant vector, capturing a net effect. This operation simply involves adding the corresponding components of each vector. From our exercise, suppose you have \( 2\mathbf{u} = \langle 2, -6, 4 \rangle \) and you wish to add (rather subtract, thereby reversing the direction) the result of \( \mathbf{v} - \mathbf{w} = \langle -3, -5, -8 \rangle \):

• First component: \( 2 + 3 = 5 \)
• Second component: \( -6 + 5 = -1 \)
• Third component: \( 4 + 8 = 12 \)

The new vector is \( \langle 5, -1, 12 \rangle \).
This process highlights how movements or forces in different directions are combined, making vector addition essential in physics for determining total force, velocity, or displacement in multiple dimensions. Graphically, vector addition can be visualized by placing the tail of one vector at the head of another, providing a clear picture of the resulting action or change.