Problem 22 - University Calculus: Early Transcendentals

Question

Express each vector in the form \(\mathbf{w}=w_{1} \mathbf{i}+\) \(w_{2} \mathbf{j}+w_{3} \mathbf{k}\). \(-2 \mathbf{u}+3 \mathbf{v}\) if \(\mathbf{u}=\langle-1,0,2\rangle\) and \(\mathbf{v}=\langle 1,1,1\rangle\)

Step-by-Step Solution

Answer

The vector \(\mathbf{w}\) is \(5\mathbf{i} + 3\mathbf{j} - \mathbf{k}\).

Step 1: Vector Multiplication by Scalar

Start by multiplying vector \(\mathbf{u}\) by the scalar \(-2\). This means you multiply each component of \(\mathbf{u}\) by \(-2\). Hence, \(-2 \mathbf{u} = -2 \times \langle -1, 0, 2 \rangle = \langle 2, 0, -4 \rangle\).

Step 2: Vector Multiplication by Scalar

Now, multiply vector \(\mathbf{v}\) by the scalar \(3\). Each component of \(\mathbf{v}\) should be multiplied by \(3\). So, \(3 \mathbf{v} = 3 \times \langle 1, 1, 1 \rangle = \langle 3, 3, 3 \rangle\).

Step 3: Add the Resultant Vectors

Next, add the results from the first two steps together. This means adding each corresponding component of the vectors \(\langle 2, 0, -4 \rangle\) and \(\langle 3, 3, 3 \rangle\). Thus, \( -2 \mathbf{u} + 3 \mathbf{v} = \langle 2+3, 0+3, -4+3 \rangle = \langle 5, 3, -1 \rangle\).

Step 4: Express in Terms of Unit Vectors

Finally, express the result in terms of the unit vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\). This means \(\mathbf{w} = w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k}\), where \(\mathbf{w} = \langle 5, 3, -1 \rangle\). So, \(\mathbf{w} = 5 \mathbf{i} + 3 \mathbf{j} - \mathbf{k}\).

Key Concepts

Vector Addition

Vector addition is fundamental in vector algebra. It involves adding two or more vectors together to create a resultant vector. Each component of the vectors is summed individually. For example, if we have a vector \(-2 \mathbf{u}\) with components \(\langle 2, 0, -4 \rangle\) and a vector \(3 \mathbf{v}\) with components \(\langle 3, 3, 3 \rangle\), you add each corresponding pair of components:

• The first components: \(2 + 3 = 5\)
• The second components: \(0 + 3 = 3\)
• The third components: \(-4 + 3 = -1\)

The result of vector addition in this example is \(\langle 5, 3, -1 \rangle\). This process forms a new vector, representing the sum of the influences of vectors \(\mathbf{u}\) and \(\mathbf{v}\).
Vector addition is used in various fields, such as physics for forces and motions, and engineering for stresses and displacements. Understanding vector addition helps in combining multiple vector quantities together.

Scalar Multiplication of Vectors

Scalar multiplication of vectors is the process of multiplying a vector by a number, known as a scalar. Each component of the vector is multiplied by the scalar, scaling the vector's magnitude without changing its direction, unless the scalar is negative, which reverses its direction.
In the given example, when we multiply vector \(\mathbf{u}\) by \(-2\), we perform the following:

• \(( -2 \times -1 = 2 )\)
• \(( -2 \times 0 = 0 )\)
• \(( -2 \times 2 = -4 )\)

This results in the vector \(\langle 2, 0, -4 \rangle\).
Next, multiplying vector \(\mathbf{v}\) by \(3\) gives:

• \(( 3 \times 1 = 3 )\)
• \(( 3 \times 1 = 3 )\)
• \(( 3 \times 1 = 3 )\)

Resulting in the vector \(\langle 3, 3, 3 \rangle\).
This process is critical in physics and engineering to adjust vector magnitudes according to forces or other influential factors.

Unit Vectors

Unit vectors are specially defined vectors with a magnitude of 1. They are essential in expressing directions in vector components. Typically, unit vectors are denoted as \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) in three dimensions, which correspond to directions along the x, y, and z axes respectively.
When expressing any vector with unit vectors, it involves breaking down the vector into components that align with these directions:

• The component along the x-axis is multiplied by \(\mathbf{i}\)
• The component along the y-axis is multiplied by \(\mathbf{j}\)
• The component along the z-axis is multiplied by \(\mathbf{k}\)

In our exercise, the vector \(\langle 5, 3, -1 \rangle\) is expressed using unit vectors as \(\mathbf{w} = 5 \mathbf{i} + 3 \mathbf{j} - \mathbf{k}\).
This representation is universal across various applications in mathematics and physics, aiding in the visualization and solving of problems involving vectors.