Question

\(\mathrm{A}^{\boldsymbol{\longrightarrow}}=2 \mathrm{i} \wedge+2 \mathrm{j} \wedge-\mathrm{k} \wedge\) and \(\mathrm{B}^{\rightarrow}=2 \mathrm{i} \wedge-\mathrm{j} \wedge-2 \mathrm{k} \wedge\) Find \(3 \mathrm{~A}^{\rightarrow}-2 \mathrm{~B}^{\rightarrow}\) (A) \(2 \mathrm{i} \wedge+7 \mathrm{j} \wedge+\mathrm{k} \wedge\) (B) \(2 \mathrm{i} \wedge+8 \mathrm{j} \wedge-\mathrm{k} \wedge\) (C) \(2 \mathrm{i} \wedge+8 \mathrm{j} \wedge+\mathrm{k} \wedge\) (D) \(\mathrm{i} \wedge+7 \mathrm{j} \wedge+\mathrm{k} \wedge\)

Short answer

The short answer is: \(3\mathrm{A} - 2\mathrm{B} = 2\mathrm{i} + 8\mathrm{j} + \mathrm{k}\)

Step-by-step solution

Multiply Vector A by 3

To multiply vector A by 3, we need to multiply each of its components by 3: \(3\mathrm{A} = 3(2\mathrm{i} + 2\mathrm{j} - \mathrm{k}) = 6\mathrm{i} + 6\mathrm{j} - 3\mathrm{k}\)

Multiply Vector B by -2

To multiply vector B by -2, we need to multiply each of its components by -2: \(-2\mathrm{B} = -2(2\mathrm{i} - \mathrm{j} -2\mathrm{k}) = -4\mathrm{i} + 2\mathrm{j} + 4\mathrm{k}\)

Add the results of Step 1 and Step 2

Now, we'll add the resulting vectors from Steps 1 and 2: \(3\mathrm{A} - 2\mathrm{B} = (6\mathrm{i} + 6\mathrm{j} - 3\mathrm{k}) + (-4\mathrm{i} + 2\mathrm{j} + 4\mathrm{k})\)

Write the final result

Now, we combine the corresponding components in the expression above: \(3\mathrm{A} - 2\mathrm{B} = (6\mathrm{i} - 4\mathrm{i}) + (6\mathrm{j} + 2\mathrm{j}) + (-3\mathrm{k} + 4\mathrm{k}) = 2\mathrm{i} + 8\mathrm{j} + \mathrm{k}\) So, the final result for \(3\mathrm{A} - 2\mathrm{B}\) is \(2\mathrm{i} + 8\mathrm{j} + \mathrm{k}\), and the correct answer is (C).

Key concepts

Vector Addition

When dealing with vectors, one frequently needs to combine multiple vectors to find their resultant. This process is known as vector addition. Imagine two vectors as arrows on a coordinate plane. To add them, you place the tail of the second vector at the head of the first vector. The resultant, or sum, is the vector drawn from the tail of the first vector to the head of the second vector.
Mathematically, vector addition involves adding each corresponding component of the vectors together. For example, when adding vector \(\boldsymbol{A = 2\hat{i} + 2\hat{j} - \hat{k}}\) and vector \(\boldsymbol{B = 2\hat{i} - \hat{j} - 2\hat{k}}\), you combine like terms:

• Combine the i-components: \(\boldsymbol{2\hat{i} + 2\hat{i} = 4\hat{i}}\).
• Combine the j-components: \(\boldsymbol{2\hat{j} - \hat{j} = \hat{j}}\).
• Combine the k-components: \(\boldsymbol{-\hat{k} - 2\hat{k} = -3\hat{k}}\).

This process simplifies vector calculations like those often encountered in physics and engineering problems.

Vector Multiplication

Vector multiplication can refer to several different operations, two of which are dot product and scalar multiplication. Scalar multiplication involves multiplying a vector by a real number (or a scalar), while dot product results in a scalar from two vectors.
In the example exercise, scalar multiplication was used to transform vector \(\boldsymbol{A}\) by 3 and vector \(\boldsymbol{B}\) by -2. This means multiplying each component of the vector by the scalar value:

• For \(\boldsymbol{A} = 2\hat{i} + 2\hat{j} - \hat{k}\), multiplying by 3 gives \(\boldsymbol{6\hat{i} + 6\hat{j} - 3\hat{k}}\).
• For \(\boldsymbol{B} = 2\hat{i} - \hat{j} - 2\hat{k}\), multiplying by -2 gives \(\boldsymbol{-4\hat{i} + 2\hat{j} + 4\hat{k}}\).

Vector multiplication is crucial for solving problems involving scaling or directions in physics and engineering.

Coordinate System

In any vector operation, understanding the coordinate system is vital. Coordinate systems allow for the positioning of vectors within a framework that provides dimensional context. Most commonly, vectors are expressed in a Cartesian coordinate system, using components along the x, y, and z axes.
In 3D space, like in the given exercise, vectors are expressed using unit vectors, often symbolized as \(\boldsymbol{\hat{i}}\), \(\boldsymbol{\hat{j}}\), and \(\boldsymbol{\hat{k}}\) which represent vectors of length one along the x, y, and z-axes respectively. A vector such as \(\boldsymbol{A = 2\hat{i} + 2\hat{j} - \hat{k}}\) indicates:

• A movement of 2 units in the x-direction (\(\boldsymbol{\hat{i}}\)).
• A movement of 2 units in the y-direction (\(\boldsymbol{\hat{j}}\)).
• A movement of -1 unit in the z-direction (\(\boldsymbol{-\hat{k}}\)).

Understanding how these unit vectors operate within the coordinate system is essential when performing vector operations.

Physics Problem Solving

In physics, vectors play a crucial role in describing quantities having both magnitude and direction, like force, velocity, and acceleration. Problem solving in physics often involves understanding and manipulating these vectors to find solutions.
The given exercise is an example of a common physics problem where you calculate a linear combination of vectors. This requires both scalar multiplication and addition of the vectors. Such skills are regularly employed in topics like mechanics, where you might need to find the net force acting on a body by combining multiple force vectors.
The steps of solving vector problems can typically include:

• Diagramming the vectors to visualize the problem.
• Performing operations such as multiplication of vectors by scalars to adjust their magnitude.
• Adding vectors to find a resultant vector.
• Interpreting the results in the context of the problem.

These methods reflect how vector operations are foundational in effectively solving a wide range of physics problems.