Problem 15 - Essential Calculus Early Transcendentals

Question

Find \(\mathbf{a}+\mathbf{b}, 2 \mathbf{a}+3 \mathbf{b},|\mathbf{a}|,\) and \(|\mathbf{a}-\mathbf{b}|\) \(\mathbf{a}=\mathbf{i}+2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{b}=-2 \mathbf{i}-\mathbf{j}+5 \mathbf{k}\)

Step-by-Step Solution

Answer

\(-\mathbf{i} + \mathbf{j} + 2\mathbf{k}, -4\mathbf{i} + \mathbf{j} + 9\mathbf{k}, \sqrt{14}, \sqrt{82}\).

Step 1: Find \(\mathbf{a} + \mathbf{b}\)

Calculate the vector sum by adding components:\(\mathbf{a} + \mathbf{b} = (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) + (-2\mathbf{i} - \mathbf{j} + 5\mathbf{k})\)Simplify each component:\(\mathbf{a} + \mathbf{b} = (1 - 2)\mathbf{i} + (2 - 1)\mathbf{j} + (-3 + 5)\mathbf{k} = -\mathbf{i} + \mathbf{j} + 2\mathbf{k}\)

Step 2: Find \(2\mathbf{a} + 3\mathbf{b}\)

First, multiply each component of \(\mathbf{a}\) and \(\mathbf{b}\) by their respective scalars:\(2\mathbf{a} = 2(\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) = 2\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\)\(3\mathbf{b} = 3(-2\mathbf{i} - \mathbf{j} + 5\mathbf{k}) = -6\mathbf{i} - 3\mathbf{j} + 15\mathbf{k}\)Now add the results:\(2\mathbf{a} + 3\mathbf{b} = (2 - 6)\mathbf{i} + (4 - 3)\mathbf{j} + (-6 + 15)\mathbf{k} = -4\mathbf{i} + \mathbf{j} + 9\mathbf{k}\)

Step 3: Find \(|\mathbf{a}|\)

Calculate the magnitude of \(\mathbf{a}\):\(|\mathbf{a}| = \sqrt{1^2 + 2^2 + (-3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14}\)

Step 4: Find \(|\mathbf{a} - \mathbf{b}|\)

First, find \(\mathbf{a} - \mathbf{b}\):\(\mathbf{a} - \mathbf{b} = (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) - (-2\mathbf{i} - \mathbf{j} + 5\mathbf{k})\)Simplify each component:\(\mathbf{a} - \mathbf{b} = (1 + 2)\mathbf{i} + (2 + 1)\mathbf{j} + (-3 - 5)\mathbf{k} = 3\mathbf{i} + 3\mathbf{j} - 8\mathbf{k}\)Now find the magnitude:\(|\mathbf{a} - \mathbf{b}| = \sqrt{3^2 + 3^2 + (-8)^2} = \sqrt{9 + 9 + 64} = \sqrt{82}\)

Key Concepts

Vector Addition

Vector addition is one of the basic operations you can perform with vectors. It involves combining two or more vectors to produce another vector. Think of each vector as an arrow pointing in space with both direction and magnitude (or length). When you add vectors, you align them head-to-tail and draw the resultant vector from the tail of the first to the head of the last. This forms a new vector, which shows the total effect of the vectors combined.

For example, consider two vectors, \( \mathbf{a} = \mathbf{i} + 2 \mathbf{j} - 3 \mathbf{k} \) and \( \mathbf{b} = -2 \mathbf{i} - \mathbf{j} + 5 \mathbf{k} \). Vector addition involves adding their corresponding components, which in this case, gives you \( \mathbf{a} + \mathbf{b} = (1 - 2)\mathbf{i} + (2 - 1)\mathbf{j} + (-3 + 5)\mathbf{k} = -\mathbf{i} + \mathbf{j} + 2\mathbf{k} \).

• Add the \( i \), \( j \), and \( k \) components separately.
• Keep the components' operations simple and straightforward.
• Ensure to maintain the sign of each component during addition.

Vector addition is not only about calculating numbers; it physically represents adding spatial directions, which is fundamental in diverse fields like physics and engineering.

Scalar Multiplication

Scalar multiplication is where you multiply a vector by a scalar (a regular number). The result is a vector whose direction is the same as the original vector, but its magnitude is scaled by the scalar. This operation is akin to stretching or shrinking the vector.

If we have a vector \( \mathbf{a} = \mathbf{i} + 2 \mathbf{j} - 3 \mathbf{k} \) and want to find \( 2\mathbf{a} \), each component of \( \mathbf{a} \) is multiplied by 2, leading to \( 2 \mathbf{a} = 2 (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) = 2\mathbf{i} + 4\mathbf{j} - 6\mathbf{k} \).

When you multiply vector \( \mathbf{b} = -2 \mathbf{i} - \mathbf{j} + 5 \mathbf{k} \) by 3, it becomes \( 3\mathbf{b} = 3(-2 \mathbf{i} - \mathbf{j} + 5 \mathbf{k}) = -6\mathbf{i} - 3\mathbf{j} + 15\mathbf{k} \).

• Take care of signs when doing multiplication.
• Remember the direction of the vector remains unaffected by the scalar; only its length changes.

Scalar multiplication is a fundamental concept used to adjust vectors either to fit certain conditions or to facilitate vector addition and other operations.

Magnitude of Vectors

The magnitude of a vector is essentially its length, which is derived from its components. It is a measure of how "long" or "strong" a vector is and is calculated using the Pythagorean theorem in three dimensions.

For a vector \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), the magnitude \(|\mathbf{a}|\) is found by \[ |\mathbf{a}| = \sqrt{1^2 + 2^2 + (-3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]. Magnitude is always a non-negative number.

• Magnitude describes how far the vector reaches.
• It's like measuring the straight-line distance of the vector tail to its head.
• The formula for magnitude involves squaring each component, summing these squares, and finally taking the square root.

This concept is crucial when you need to normalize vectors or when determining distances in vector spaces.

Vector Subtraction

Vector subtraction involves taking the difference between two vectors. It's like reversing the direction of the vector you are subtracting and then adding it to the initial vector. Vector subtraction is used to determine the change from one vector to another.

To find the difference \( \mathbf{a} - \mathbf{b} \) for vectors \( \mathbf{a} = \mathbf{i} + 2 \mathbf{j} - 3 \mathbf{k} \) and \( \mathbf{b} = -2 \mathbf{i} - \mathbf{j} + 5 \mathbf{k} \), you subtract corresponding components: \( \mathbf{a} - \mathbf{b} = (1 + 2)\mathbf{i} + (2 + 1)\mathbf{j} + (-3 - 5)\mathbf{k} = 3\mathbf{i} + 3\mathbf{j} - 8\mathbf{k} \).

Finally, finding the magnitude of this resulting vector involves calculating \( |\mathbf{a} - \mathbf{b}| = \sqrt{3^2 + 3^2 + (-8)^2} = \sqrt{9 + 9 + 64} = \sqrt{82} \).

• Consider the sign changes when subtracting components.
• Vector subtraction gives insight into relative positions and directions.
• The result is a vector that effectively shows the direction and distance from the head of \( \mathbf{b} \) to the head of \( \mathbf{a} \).

Understanding vector subtraction helps in analyzing directional changes and is widely used in physics, engineering, and computer graphics.