Problem 9 - Calculus: Early Transcendentals

Question

Find \(|\boldsymbol{v}|, \boldsymbol{v}+\boldsymbol{w}, \boldsymbol{v}-\boldsymbol{w},|\boldsymbol{v}+\boldsymbol{w}|,|\boldsymbol{v}-\boldsymbol{w}|\) and \(-2 \boldsymbol{v}\) for \(\boldsymbol{v}=\langle 1,-1,1\rangle\) and \(\boldsymbol{w}=\langle 0,0,3\rangle .\)

Step-by-Step Solution

Answer

\(|\boldsymbol{v}| = \sqrt{3}, \boldsymbol{v} + \boldsymbol{w} = \langle 1, -1, 4 \rangle, \boldsymbol{v} - \boldsymbol{w} = \langle 1, -1, -2 \rangle, |\boldsymbol{v} + \boldsymbol{w}| = \sqrt{18}, |\boldsymbol{v} - \boldsymbol{w}| = \sqrt{6}, -2 \boldsymbol{v} = \langle -2, 2, -2 \rangle.\)

Step 1: Find Magnitude of Vector \( \boldsymbol{v} \)

To find the magnitude of a vector \( \boldsymbol{v} = \langle a, b, c \rangle \), use the formula: \( |\boldsymbol{v}| = \sqrt{a^2 + b^2 + c^2} \). For \( \boldsymbol{v} = \langle 1, -1, 1 \rangle \), calculate: \[ |\boldsymbol{v}| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{1 + 1 + 1} = \sqrt{3}. \]

Step 2: Compute \( \boldsymbol{v} + \boldsymbol{w} \)

To add vectors \( \boldsymbol{v} = \langle 1, -1, 1 \rangle \) and \( \boldsymbol{w} = \langle 0, 0, 3 \rangle \), add the corresponding components: \[ \boldsymbol{v} + \boldsymbol{w} = \langle 1 + 0, -1 + 0, 1 + 3 \rangle = \langle 1, -1, 4 \rangle. \]

Step 3: Compute \( \boldsymbol{v} - \boldsymbol{w} \)

To subtract vectors \( \boldsymbol{w} \) from \( \boldsymbol{v} \), subtract the corresponding components: \[ \boldsymbol{v} - \boldsymbol{w} = \langle 1 - 0, -1 - 0, 1 - 3 \rangle = \langle 1, -1, -2 \rangle. \]

Step 4: Find Magnitude of \( \boldsymbol{v} + \boldsymbol{w} \)

Use the formula for magnitude on the vector \( \boldsymbol{v} + \boldsymbol{w} = \langle 1, -1, 4 \rangle \): \[ |\boldsymbol{v} + \boldsymbol{w}| = \sqrt{1^2 + (-1)^2 + 4^2} = \sqrt{1 + 1 + 16} = \sqrt{18}. \]

Step 5: Find Magnitude of \( \boldsymbol{v} - \boldsymbol{w} \)

Now find the magnitude of \( \boldsymbol{v} - \boldsymbol{w} = \langle 1, -1, -2 \rangle \): \[ |\boldsymbol{v} - \boldsymbol{w}| = \sqrt{1^2 + (-1)^2 + (-2)^2} = \sqrt{1 + 1 + 4} = \sqrt{6}. \]

Step 6: Compute \( -2 \boldsymbol{v} \)

To find \( -2 \boldsymbol{v} \), multiply each component of \( \boldsymbol{v} = \langle 1, -1, 1 \rangle \) by \(-2\): \[ -2 \boldsymbol{v} = \langle -2 \times 1, -2 \times (-1), -2 \times 1 \rangle = \langle -2, 2, -2 \rangle. \]

Key Concepts

Magnitude of a Vector

The magnitude of a vector, often referred to as the length or norm, is a measure of the size of the vector. It is calculated as the square root of the sum of the squares of its components. This may sound complicated, but it's quite straightforward!
Imagine a vector \( \boldsymbol{v} = \langle a, b, c \rangle \). The formula to find the magnitude, \( |\boldsymbol{v}| \), is given by:

• Sum of the squares: \( a^2 + b^2 + c^2 \)
• Take the square root of that sum: \( \sqrt{a^2 + b^2 + c^2} \)

This gives you a single positive number representing the vector's magnitude. Let’s examine an example. For \( \boldsymbol{v} = \langle 1, -1, 1 \rangle \), you calculate: \[ |\boldsymbol{v}| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{3} .\]
This value represents how long the vector \( \boldsymbol{v} \) is in space.

Vector Addition

Vector addition is the process of adding two vectors together to create a new vector. This is done by adding their corresponding components. Remember, you deal with vectors component-wise!
Let's add two vectors, \( \boldsymbol{v} = \langle 1, -1, 1 \rangle \) and \( \boldsymbol{w} = \langle 0, 0, 3 \rangle \):

• Add the x-components: \( 1 + 0 = 1 \)
• Add the y-components: \( -1 + 0 = -1 \)
• Add the z-components: \( 1 + 3 = 4 \)

Thus, the resulting vector is \( \boldsymbol{v} + \boldsymbol{w} = \langle 1, -1, 4 \rangle \).
It’s like combining forces or paths, where one vector starts where the other ends.

Vector Subtraction

Vector subtraction is similar to addition but involves subtracting corresponding components to find the difference between two vectors. Think of it as finding out what vector you add to one vector to reach the other.
For vectors \( \boldsymbol{v} = \langle 1, -1, 1 \rangle \) and \( \boldsymbol{w} = \langle 0, 0, 3 \rangle \), find \( \boldsymbol{v} - \boldsymbol{w} \):

• Subtract the x-components: \( 1 - 0 = 1 \)
• Subtract the y-components: \( -1 - 0 = -1 \)
• Subtract the z-components: \( 1 - 3 = -2 \)

So, \( \boldsymbol{v} - \boldsymbol{w} = \langle 1, -1, -2 \rangle \).
You can picture vector subtraction as reversing one vector and then adding it to the first.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying every component of a vector by a scalar (a constant number). This operation stretches or shrinks the vector and can also reverse its direction if the scalar is negative.
For the vector \( \boldsymbol{v} = \langle 1, -1, 1 \rangle \) and scalar \(-2\):

• Multiply the x-component: \(-2 \times 1 = -2 \)
• Multiply the y-component: \(-2 \times (-1) = 2 \)
• Multiply the z-component: \(-2 \times 1 = -2 \)

Thus, the result is \( -2 \boldsymbol{v} = \langle -2, 2, -2 \rangle \).
This operation effectively doubles the vector’s magnitude and reverses its direction since the scalar is \(-2\).