Problem 72
Question
Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\). $$\mathbf{v}=2(\mathbf{u}-\mathbf{w})$$
Step-by-Step Solution
Verified Answer
The component form of the vector \(v\) is \(v= 2\mathbf{i} - 6\mathbf{j}\)
1Step 1: Understand Vector Subtraction
Subtract vector \(\mathbf{w}\) from vector \(\mathbf{u}\). To do this, subtract corresponding \(\mathbf{i}\) components and corresponding \(\mathbf{j}\) components of \(\mathbf{u}\) and \(\mathbf{w}\) from each other. Hence we get a new vector, say \(\mathbf{b}\), where \(\mathbf{b} = \mathbf{u} - \mathbf{w} = (2\mathbf{i}-\mathbf{j}) - (\mathbf{i}+2 \mathbf{j}) = \mathbf{i} - 3\mathbf{j}\).
2Step 2: Scalar multiplication
Multiply the resultant vector \(\mathbf{b}\) by 2. This operation results in the vector \(\mathbf{v}\), where \(\mathbf{v} = 2 \cdot \mathbf{b} = 2(\mathbf{i}-3\mathbf{j}) = 2\mathbf{i} - 6\mathbf{j}\). This is the component form of the vector \(\mathbf{v}\).
3Step 3: Geometric Representation
Geometrically, the vectors \(\mathbf{u}\), \(\mathbf{w}\), and \(\mathbf{v}\) can be represented in a two-dimensional Cartesian system where î is along the x-axis and ĵ is along the y-axis. The vector \(\mathbf{u}\) can be represented as a line segment from the origin (0,0) to the point (2,-1), \(\mathbf{w}\) from (0,0) to (1,2) and our resultant vector \(\mathbf{v}\) from (0,0) to (2,-6).
Key Concepts
Component FormScalar MultiplicationVector Subtraction
Component Form
Vectors in component form are expressed using their horizontal and vertical influences in a given coordinate system. Imagine a vector as a journey through a grid: it tells you how far to travel horizontally and vertically. This is done using unit vectors, typically \(\mathbf{i}\) and \(\mathbf{j}\).
For example, the vector \(\mathbf{v}\) is given in terms of \(\mathbf{i}\) and \(\mathbf{j}\) as \(2\mathbf{i} - 6\mathbf{j}\). This means you move 2 units right (horizontal) and 6 units down (vertical). This form helps us easily perform calculations involving vectors, such as finding their length or direction.
To convert from geometric representation to component form, simply use the horizontal and vertical distances (or projections) from the start to the endpoint of the vector. Working with component form is intuitive and simplifies vector operations like addition, subtraction, and multiplication. Practicing this concept will make these operations second nature.
For example, the vector \(\mathbf{v}\) is given in terms of \(\mathbf{i}\) and \(\mathbf{j}\) as \(2\mathbf{i} - 6\mathbf{j}\). This means you move 2 units right (horizontal) and 6 units down (vertical). This form helps us easily perform calculations involving vectors, such as finding their length or direction.
To convert from geometric representation to component form, simply use the horizontal and vertical distances (or projections) from the start to the endpoint of the vector. Working with component form is intuitive and simplifies vector operations like addition, subtraction, and multiplication. Practicing this concept will make these operations second nature.
Scalar Multiplication
Scalar multiplication involves changing the length or magnitude of a vector by a constant factor known as a scalar. It's like adjusting the volume of a song to be louder or quieter, but here, it's about making the vector longer or shorter.
In mathematical terms, if you have a vector \(\mathbf{b} = \mathbf{i} - 3\mathbf{j}\) and a scalar 2, scalar multiplication results in \(2\mathbf{b} = 2\cdot (\mathbf{i} - 3\mathbf{j})=2\mathbf{i} - 6\mathbf{j}\). This operation widens the vector's reach along the same direction by the factor of 2. Hence, the vector not only maintains its direction but also has double the original magnitude.
Scalar multiplication is one of the fundamental operations in vector algebra because it allows manipulation of vector size while maintaining its essential directionality. This operation is widely used in physics and engineering to model forces, velocity, and other vector quantities magnified or reduced by specific factors.
In mathematical terms, if you have a vector \(\mathbf{b} = \mathbf{i} - 3\mathbf{j}\) and a scalar 2, scalar multiplication results in \(2\mathbf{b} = 2\cdot (\mathbf{i} - 3\mathbf{j})=2\mathbf{i} - 6\mathbf{j}\). This operation widens the vector's reach along the same direction by the factor of 2. Hence, the vector not only maintains its direction but also has double the original magnitude.
Scalar multiplication is one of the fundamental operations in vector algebra because it allows manipulation of vector size while maintaining its essential directionality. This operation is widely used in physics and engineering to model forces, velocity, and other vector quantities magnified or reduced by specific factors.
Vector Subtraction
Subtracting one vector from another is like finding the effective movement between two positions. Imagine both vectors as routes; subtracting finds a direct shortcut from one's endpoint back to the other's.
For instance, given \(\mathbf{u}=2\mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2\mathbf{j}\), subtract \(\mathbf{w}\) from \(\mathbf{u}\) by subtracting their corresponding components: \(2\mathbf{i} - \mathbf{j} - (\mathbf{i} + 2\mathbf{j}) = \mathbf{i} - 3\mathbf{j}\).
This subtraction provides a new vector, representing the net movement directly from \(\mathbf{w}\) to \(\mathbf{u}\). It's equivalent to reversing one vector and adding it to the other.
Vector subtraction is crucial in scenarios requiring calculation of relative positions or changes, such as determining displacement, differential velocity, and other comparative measures in physics and math.
For instance, given \(\mathbf{u}=2\mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2\mathbf{j}\), subtract \(\mathbf{w}\) from \(\mathbf{u}\) by subtracting their corresponding components: \(2\mathbf{i} - \mathbf{j} - (\mathbf{i} + 2\mathbf{j}) = \mathbf{i} - 3\mathbf{j}\).
This subtraction provides a new vector, representing the net movement directly from \(\mathbf{w}\) to \(\mathbf{u}\). It's equivalent to reversing one vector and adding it to the other.
Vector subtraction is crucial in scenarios requiring calculation of relative positions or changes, such as determining displacement, differential velocity, and other comparative measures in physics and math.
Other exercises in this chapter
Problem 71
Perform the operation and leave the result in trigonometric form. $$\frac{\cos 50^{\circ}+i \sin 50^{\circ}}{\cos 20^{\circ}+i \sin 20^{\circ}}$$
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A truck with a gross weight of 30,000 pounds is parked on a slope of \(d^{\circ}\) (see figure). Assume that the only force to overcome is the force of gravity.
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