Problem 31 - Thomas Calculus

Question

Find the vectors whose lengths and directions are given. Try to do the calculations without writing. Length \(\quad\) Direction a. 2\(\quad\quad\) i b. \(\sqrt{3} \quad\quad-\mathbf{k}\) c. \(\frac{1}{2} \quad\quad \frac{3}{5} \mathbf{j}+\frac{4}{5} \mathbf{k}\) d. \(7 \quad \quad \frac{6}{7} \mathbf{i}-\frac{2}{7} \mathbf{j}+\frac{3}{7} \mathbf{k}\)

Step-by-Step Solution

Answer

a. \(2\mathbf{i}\); b. \(-\sqrt{3}\mathbf{k}\); c. \(\frac{3}{10}\mathbf{j} + \frac{2}{5}\mathbf{k}\); d. \(6\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}\)."

Step 1: Understanding the Length and Direction

The vector is defined by its length and direction. The direction is often expressed as a unit vector. To find the vector, multiply the unit vector by the given length.

Step 2: Calculating Vector a

For vector a, the length is 2, and the direction is along the unit vector \( \mathbf{i} \). So, the vector is \( 2 \mathbf{i} \).

Step 3: Calculating Vector b

For vector b, the length is \( \sqrt{3} \), and the direction is opposite to unit vector \( \mathbf{k} \) (i.e., \( -\mathbf{k} \)). So, the vector is \( -\sqrt{3} \mathbf{k} \).

Step 4: Identifying the Unit Vector for Vector c

The unit vector direction is \( \frac{3}{5} \mathbf{j} + \frac{4}{5} \mathbf{k} \). This is a unit vector since its magnitude is 1: \( \sqrt{\left( \frac{3}{5} \right)^2 + \left( \frac{4}{5} \right)^2} = 1 \).

Step 5: Calculating Vector c

Multiply the unit vector by the length \( \frac{1}{2} \), giving the vector \( \frac{1}{2} \left( \frac{3}{5} \mathbf{j} + \frac{4}{5} \mathbf{k} \right) = \frac{3}{10} \mathbf{j} + \frac{2}{5} \mathbf{k} \).

Step 6: Identifying the Unit Vector for Vector d

The unit vector direction for vector d is \( \frac{6}{7} \mathbf{i} - \frac{2}{7} \mathbf{j} + \frac{3}{7} \mathbf{k} \). This is a unit vector, as it also has a magnitude of 1.

Step 7: Calculating Vector d

Multiply the unit vector by the length 7, getting the vector \( 7 \left( \frac{6}{7} \mathbf{i} - \frac{2}{7} \mathbf{j} + \frac{3}{7} \mathbf{k} \right) = 6 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k} \).

Key Concepts

Unit Vector

A unit vector is a vector that has a magnitude of 1. Unit vectors are used to indicate the direction of a vector. They are often denoted with a hat (^) on top of the vector symbol, such as \( \hat{i} \), \( \hat{j} \), or \( \hat{k} \). This indicates they maintain direction but their length is precisely one unit.
In calculations, you can express any vector direction as a unit vector by dividing the vector by its magnitude. This ensures the resultant vector keeps the direction but becomes a unit vector. For example, if a direction is given as \( \frac{3}{5} \mathbf{j} + \frac{4}{5} \mathbf{k} \), it already represents a unit vector since its magnitude is 1. Such a vector can then simply be scaled to any desired length by multiplying with the required magnitude.

Vector Length

The vector length, also known as the magnitude of a vector, is a measure of how long the vector is. It is always a non-negative value and is denoted by \( \| \mathbf{v} \| \) for a vector \( \mathbf{v} \). The length of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) is calculated using the formula:

• \( \sqrt{a^2 + b^2 + c^2} \)

The length helps in understanding the magnitude of a physical quantity represented by the vector, such as displacement or force.
In exercises like the one above, when given a specific length and unit vector, you multiply the unit vector by the length to find the final vector, ensuring it maintains the intended magnitude alongside the direction.

Vector Direction

Vector direction describes the orientation of the vector in space. In three-dimensional space, direction is determined by how much the vector points along each axis, typically represented using components in the \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) directions.
To determine vector direction, you take the components of the vector and analyze their relative values and signs. For example, a vector directed as \( \frac{6}{7} \mathbf{i} - \frac{2}{7} \mathbf{j} + \frac{3}{7} \mathbf{k} \) means it leans more towards the positive x-axis while slightly in the direction of the negative y-axis and somewhat towards the positive z-axis.
Understanding direction is crucial for obtaining the correct vector orientation in many physics and engineering applications, guiding the direction in which forces are applied or objects move.

Magnitude of Vector

The magnitude of a vector is the "size" or "length" of the vector and is a scalar quantity, meaning it only has magnitude and no direction. It's calculated using the formula mentioned earlier:

• \( \sqrt{a^2 + b^2 + c^2} \)

where \( a \), \( b \), and \( c \) are the vector's components in the \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) directions, respectively.
The magnitude of a vector helps identify how much of a quantity is being represented, much like how a distance might represent the space between two points. Even when a vector direction changes, its magnitude might remain constant, reflecting only the strength of the vector's influence, such as force magnitude remaining the same when applied in different directions.