Question

Two vectors are given by \(\overrightarrow{\mathbf{a}}=4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{b}}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}+4 \hat{\mathbf{k}} .\) Find \((a) \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}},(b) \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}\), and \((c)\) a vector \(\overrightarrow{\mathbf{c}}\) such that \(\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=0\)

Short answer

The sum of vectors \( \overrightarrow{\mathbf{a}} \) and \( \overrightarrow{\mathbf{b}} \) is \( 3\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 5\hat{\mathbf{k}} \). The difference of vectors \( \overrightarrow{\mathbf{a}} \) and \( \overrightarrow{\mathbf{b}} \) is \( 5\hat{\mathbf{i}}-4\hat{\mathbf{j}}-3\hat{\mathbf{k}} \). The vector \( \overrightarrow{\mathbf{c}} \) that satisfies the condition \( \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=0 \) is \( -5\hat{\mathbf{i}} + 4\hat{\mathbf{j}} + 3 \hat{\mathbf{k}} \).

Step-by-step solution

Adding Vectors

Vectors are added by component. So to find \( \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} \), the i, j, and k components of the two vectors should be added separately. \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} = (4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}) + (-\hat{\mathbf{i}}+\hat{\mathbf{j}}+4 \hat{\mathbf{k}}) = 3\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 5\hat{\mathbf{k}} \)

Subtracting Vectors

Like addition, vector subtraction is also done by subtracting corresponding components. So to find \( \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}} \), subtract the i, j, and k components of vector b from vector a. \(\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}} = (4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}) - (-\hat{\mathbf{i}}+\hat{\mathbf{j}}+4 \hat{\mathbf{k}}) = 5\hat{\mathbf{i}} - 4\hat{\mathbf{j}} - 3\hat{\mathbf{k}} \)

Find Vector \( \overrightarrow{\mathbf{c}} \)

To make the equation \( \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=0 \) correct, \( \overrightarrow{\mathbf{c}} \) should be equal to \( -\overrightarrow{\mathbf{a}} +\overrightarrow{\mathbf{b}} \).We find it by subtracting vector a from vector b: \( \overrightarrow{\mathbf{c}} = -\overrightarrow{\mathbf{a}} +\overrightarrow{\mathbf{b}} = -4\hat{\mathbf{i}} +3\hat{\mathbf{j}} -\hat{\mathbf{k}} + -\hat{\mathbf{i}}+\hat{\mathbf{j}}+4 \hat{\mathbf{k}} = -5\hat{\mathbf{i}} + 4\hat{\mathbf{j}} + 3 \hat{\mathbf{k}} \).

Key concepts

Vector Addition

Vector addition is a crucial concept in mathematics, particularly useful in physics and engineering. Think of vectors as arrows, each with a direction and magnitude.
Adding vectors involves combining these arrows to find a resulting vector.

• To add vectors, you simply add their corresponding components.
• This means adding the i-components (horizontal), j-components (vertical), and k-components (depth, in 3D) of the vectors.
• The resulting vector is a new vector that shows the combined effect of these initial vectors.

For example, given vectors \( \overrightarrow{\mathbf{a}} = 4 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + \hat{\mathbf{k}} \) and \( \overrightarrow{\mathbf{b}} = -\hat{\mathbf{i}} + \hat{\mathbf{j}} + 4 \hat{\mathbf{k}} \), adding them results in a new vector: \[\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} = (4 - 1)\hat{\mathbf{i}} + (-3 + 1)\hat{\mathbf{j}} + (1 + 4)\hat{\mathbf{k}} = 3\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 5\hat{\mathbf{k}}.\]So, by adding these vectors, you find a vector pointing in the direction of \( 3\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 5\hat{\mathbf{k}} \). This represents the combination of the effects of \( \overrightarrow{\mathbf{a}} \) and \( \overrightarrow{\mathbf{b}} \).
Understanding vector addition is foundational for working with forces, velocities, and more in multi-dimensional spaces.

Vector Subtraction

Vector subtraction is very similar to vector addition, but instead of adding the components, you subtract them. Visualize this as removing or reversing the direction of one vector while combining it with another.

• To subtract vectors, subtract the components of the second vector from those of the first vector.
• This calculation is done component-wise: \( i \) from \( i \), \( j \) from \( j \), and \( k \) from \( k \).

For instance, with vectors \( \overrightarrow{\mathbf{a}} = 4 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + \hat{\mathbf{k}} \) and \( \overrightarrow{\mathbf{b}} = -\hat{\mathbf{i}} + \hat{\mathbf{j}} + 4 \hat{\mathbf{k}} \), subtracting them gives:\[ \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}} = (4 + 1)\hat{\mathbf{i}} + (-3 - 1)\hat{\mathbf{j}} + (1 - 4)\hat{\mathbf{k}} = 5\hat{\mathbf{i}} - 4\hat{\mathbf{j}} - 3\hat{\mathbf{k}}.\]Hence, the resulting vector \( 5\hat{\mathbf{i}} - 4\hat{\mathbf{j}} - 3\hat{\mathbf{k}} \) indicates the direction and magnitude after the effects of vector \( \overrightarrow{\mathbf{b}} \) have been removed from vector \( \overrightarrow{\mathbf{a}} \).
Mastering vector subtraction allows you to analyze scenarios like finding relative positions or determining net effects.

Vector Components

Vectors are broken down into components to simplify their manipulation and understanding. Imagine each vector as a combination of basic vectors along the coordinate axes.

• Most commonly, vectors are expressed in terms of \( \hat{\mathbf{i}} \), \( \hat{\mathbf{j}} \), and \( \hat{\mathbf{k}} \) which represent unit vectors along the x, y, and z axes respectively.
• These components allow you to easily break a vector into its effects in each dimension.

For any vector \( \overrightarrow{\mathbf{v}} = a\hat{\mathbf{i}} + b\hat{\mathbf{j}} + c\hat{\mathbf{k}} \): - \( a \) is the scalar component in the x-direction,- \( b \) is the scalar component in the y-direction,- \( c \) is the scalar component in the z-direction.
Knowing how to interpret vector components is essential for vector addition and subtraction. They make it possible to visually and mathematically evaluate how vectors interact in different directions. This forms the basis for more complex studies in physics, such as analyzing forces or velocities in multiple dimensions.
By focusing on these individual components, you can logically and easily deal with otherwise complex multi-dimensional problems.