Problem 8 - 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 \(\boldsymbol{v}=\langle 1,0,1\rangle\) and \(\boldsymbol{w}=\langle-1,-2,2\rangle .\)

Step-by-Step Solution

Answer

The magnitudes are \( \sqrt{2} \), \( \sqrt{13} \), and 3. The vectors are \( \langle 0, -2, 3 \rangle \), \( \langle 2, 2, -1 \rangle \), and \( \langle -2, 0, -2 \rangle \).

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

The magnitude of a vector \( \langle a, b, c \rangle \) is calculated by the formula \( \sqrt{a^2 + b^2 + c^2} \). For \( \boldsymbol{v} = \langle 1, 0, 1 \rangle \), the magnitude is \( |\boldsymbol{v}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2} \).

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

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

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

Subtract the corresponding components of the vectors: \( \boldsymbol{v} - \boldsymbol{w} = \langle 1, 0, 1 \rangle - \langle -1, -2, 2 \rangle = \langle 2, 2, -1 \rangle \).

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

Use the formula for magnitude on \( \boldsymbol{v} + \boldsymbol{w} = \langle 0, -2, 3 \rangle \): \( |\boldsymbol{v} + \boldsymbol{w}| = \sqrt{0^2 + (-2)^2 + 3^2} = \sqrt{13} \).

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

Calculate the magnitude of \( \boldsymbol{v} - \boldsymbol{w} = \langle 2, 2, -1 \rangle \): \( |\boldsymbol{v} - \boldsymbol{w}| = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{9} = 3 \).

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

Multiply each component of \( \boldsymbol{v} \) by \(-2\): \(-2\boldsymbol{v} = -2\langle 1, 0, 1 \rangle = \langle -2, 0, -2 \rangle \).

Key Concepts

Vector Addition

Adding vectors is like combining forces. You take two vectors, in this case \( \boldsymbol{v} = \langle 1, 0, 1 \rangle \) and \( \boldsymbol{w} = \langle -1, -2, 2 \rangle \), and create a new vector by summing their components. That means:

• Add the first components: \( 1 + (-1) = 0 \)
• Add the second components: \( 0 + (-2) = -2 \)
• Add the third components: \( 1 + 2 = 3 \)

Thus, the resulting vector from the addition, \( \boldsymbol{v} + \boldsymbol{w} \), is \( \langle 0, -2, 3 \rangle \). Think of it as simply moving in a new direction that results from both original paths.

Vector Subtraction

Subtracting vectors initially might sound a bit intimidating, but it's straightforward. Vector subtraction entails subtracting each corresponding component of one vector from another. For \( \boldsymbol{v} = \langle 1, 0, 1 \rangle \) and \( \boldsymbol{w} = \langle -1, -2, 2 \rangle \):

• Subtract the first components: \( 1 - (-1) = 2 \)
• Subtract the second components: \( 0 - (-2) = 2 \)
• Subtract the third components: \( 1 - 2 = -1 \)

The resulting vector, \( \boldsymbol{v} - \boldsymbol{w} \), is \( \langle 2, 2, -1 \rangle \). It indicates the relative position change if you were moving in the direction of \( \boldsymbol{v} \) while simultaneously reversing direction from \( \boldsymbol{w} \).

Magnitude of a Vector

The magnitude of a vector gives us its length or how far it extends from the origin, serving as a measure of its size. Calculating this involves the Pythagorean theorem and finding the square root of the sum of squared components. For the vector \( \boldsymbol{v} = \langle 1, 0, 1 \rangle \), the magnitude \( |\boldsymbol{v}| \) is:\[|\boldsymbol{v}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}\]Similarly, to find the magnitude of \( \boldsymbol{v} + \boldsymbol{w} = \langle 0, -2, 3 \rangle \), it results in:\[|\boldsymbol{v} + \boldsymbol{w}| = \sqrt{0^2 + (-2)^2 + 3^2} = \sqrt{13}\]And for \( \boldsymbol{v} - \boldsymbol{w} = \langle 2, 2, -1 \rangle \), it's measured as:\[|\boldsymbol{v} - \boldsymbol{w}| = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{9} = 3\]The magnitude quantifies how long the vector physically spans, regardless of its direction.

Scalar Multiplication

Scalar multiplication involves stretching or compressing the vector in a consistent manner. It multiplies each component of a given vector by a scalar (a real number), modifying its length but not its direction, unless the scalar is negative, which reverses the direction.For example, the vector \( \boldsymbol{v} = \langle 1, 0, 1 \rangle \) multiplied by the scalar \(-2\) results in:

• Multiply the first component: \( -2 \times 1 = -2 \)
• Multiply the second component: \( -2 \times 0 = 0 \)
• Multiply the third component: \( -2 \times 1 = -2 \)

Thus, \(-2\boldsymbol{v}\) becomes \( \langle -2, 0, -2 \rangle \). Notice that the vector's direction flipped since the scalar was negative. This operation scales and flips vectors appropriately, useful in various applications such as physics for adjusting force and direction.