Question

Find the modulus of each of the following vectors: (a) \(\boldsymbol{r}=7 \boldsymbol{i}+3 \boldsymbol{j}\) (b) \(\boldsymbol{r}=17 \boldsymbol{i}\) (c) \(\boldsymbol{r}=2 \boldsymbol{i}-3 \boldsymbol{j}\) (d) \(r=-3 j\) (f) \(\boldsymbol{r}=a \boldsymbol{i}-b \boldsymbol{j}\)

Short answer

Question: Calculate the moduli (magnitudes) of the following vectors: (a) \(r=7i+3j\), (b) \(r=17i\), (c) \(r=2i-3j\), (d) \(r=-3j\), and (f) \(r=a i-b j\). Answer: (a) \(\|r\| \approx 7.62\) (b) \(\|r\| = 17\) (c) \(\|r\| \approx 3.61\) (d) \(\|r\| = 3\) (f) \(\|r\| = \sqrt{(a^2+b^2)}\)

Step-by-step solution

Calculate the modulus of (a) \(r=7i+3j\)

To calculate the modulus of this vector, we will apply the formula: \(\|r\|=\sqrt{(7^2+3^2)}\). Calculate the square of each component and sum them: \((7^2 + 3^2) = 49 + 9 = 58\). Now, find the square root of the result: \(\|r\|=\sqrt{58} \approx 7.62\). So, the modulus of the vector \(r=7i+3j\) is approximately \(7.62\).

Calculate the modulus of (b) \(r=17i\)

In this case, the vector has only one component. So, the formula to calculate the modulus will be simply: \(\|r\|=\sqrt{(17^2)}\). Calculate the square of the component: \((17^2) = 289\). And find the square root of the result: \(\|r\|=\sqrt{289} = 17\). Thus, the modulus of the vector \(r=17i\) is \(17\).

Calculate the modulus of (c) \(r=2i-3j\)

To calculate the modulus of this vector, we will apply the formula: \(\|r\|=\sqrt{(2^2+(-3)^2)}\). Calculate the square of each component and sum them: \((2^2 + (-3)^2) = 4 + 9 = 13\). Now, find the square root of the result: \(\|r\|=\sqrt{13} \approx 3.61\). Thus, the modulus of the vector \(r=2i-3j\) is approximately \(3.61\).

Calculate the modulus of (d) \(r=-3j\)

In this case, the vector also has only one component. So, the formula to calculate the modulus will be simply: \(\|r\|=\sqrt{((-3)^2)}\). Calculate the square of the component: \(((-3)^2) = 9\). And find the square root of the result: \(\|r\|=\sqrt{9} = 3\). Thus, the modulus of the vector \(r=-3j\) is \(3\).

Calculate the modulus of (f) \(r=a i-b j\)

For this vector, we don't have specific values for the components. However, we can still set up the general formula to calculate its modulus: \(\|r\|=\sqrt{(a^2+b^2)}\). Thus, the modulus of the vector \(r=a i-b j\) is \(\sqrt{(a^2+b^2)}\).

Key concepts

Vector Components

Understanding vector components is essential when dealing with vector calculus. Each vector in a two-dimensional space can be decomposed into its components along the horizontal and vertical axes. These are conventionally represented by 'i' for the horizontal component and 'j' for the vertical component.

For instance, given a vector \( \boldsymbol{r} = a \boldsymbol{i} + b \boldsymbol{j} \), 'a' represents the magnitude along the 'i' or x-axis and 'b' represents the magnitude along the 'j' or y-axis. Each of these magnitudes can be positive or negative, indicating the direction of the vector in their respective axes. The real power of expressing vectors in terms of their components lies in the simplicity it brings to vector operations such as addition, subtraction, and even finding the modulus—the length of the vector.

Modulus Calculation

The modulus of a vector is a measure of its length. Mathematically, it is calculated using the principles of Pythagoras's theorem, particularly when dealing with a two-dimensional vector.

To find the modulus, which is also referred to as the magnitude or norm, we sum the squares of each component and then take the square root of this sum. The modulus of a vector \( \boldsymbol{r} = a \boldsymbol{i} + b \boldsymbol{j} \) is given by the formula \( \|r\| = \sqrt{a^2 + b^2} \). When either 'a' or 'b' is zero, the formula simplifies to the absolute value of the non-zero component, as seen in part (b) and (d) of the original exercise.

Vector Algebra

Vector algebra allows us to perform mathematical operations on vectors. Just like traditional algebra deals with scalars, vector algebra works with both magnitude and direction. It comprises various operations such as addition, subtraction, scalar multiplication, dot product, and cross product, each of which has physical and geometric interpretations.

Scalar Multiplication

When a vector is multiplied by a scalar, it results in a vector that has been scaled in magnitude, but its direction remains unchanged unless the scalar is negative, which would also reverse the direction of the vector.

Vector Addition and Subtraction

These operations are performed component-wise, demonstrating the usefulness of expressing vectors in terms of their 'i' and 'j' components. The result of addition or subtraction is a new vector that can then be manipulated or have its modulus calculated, as needed in various applications.

Euclidean Vector Norm

The Euclidean vector norm, commonly just called the norm, is a generalization of the notion of vector modulus. In a Euclidean space—essentially, a space defined by Euclidean geometry—the norm of a vector is the distance from the origin to the point represented by the vector.

This concept isn't limited to two or even three dimensions; it can be applied to an n-dimensional Euclidean space. The formula for the Euclidean norm of a vector \( \boldsymbol{r} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + ... + a_n \boldsymbol{k} \) in 'n' dimensions generalizes to \( \|r\| = \sqrt{a_1^2 + a_2^2 + ... + a_n^2} \), which is an extension of the modulus calculation in two dimensions. The profound simplicity of the Euclidean norm makes it one of the fundamental tools in vector analysis and linear algebra.