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Magazine Chapter 5: Problem 27
Sketch the vectors \(2 \mathbf{v}\) and \(-2 \mathbf{v}\). $$ \mathbf{v}=3 \mathbf{i}-5 \mathbf{j} $$
Short Answer
Expert verified
Sketch vectors at (6, -10) and (-6, 10) for 2v and -2v, respectively.
Step by step solution
01
Understand the Vector Components
The vector \( \mathbf{v} \) is given as \( 3 \mathbf{i} - 5 \mathbf{j} \). This indicates that the vector has a component of 3 in the direction of the x-axis (\( \mathbf{i} \)) and a component of -5 in the direction of the y-axis (\( \mathbf{j} \)).
02
Calculate the Vector 2v
To find \( 2 \mathbf{v} \), multiply each component of \( \mathbf{v} \) by 2. This results in \( 2 \times 3 \mathbf{i} + 2 \times (-5) \mathbf{j} = 6 \mathbf{i} - 10 \mathbf{j} \). So, \( 2 \mathbf{v} \) is \( 6 \mathbf{i} - 10 \mathbf{j} \).
03
Calculate the Vector -2v
To find \( -2 \mathbf{v} \), multiply each component of \( \mathbf{v} \) by -2. This results in \( -2 \times 3 \mathbf{i} + (-2) \times (-5) \mathbf{j} = -6 \mathbf{i} + 10 \mathbf{j} \). So, \( -2 \mathbf{v} \) is \( -6 \mathbf{i} + 10 \mathbf{j} \).
04
Sketch the Vector 2v
Draw a vector starting from the origin (0,0) to the point (6, -10) on a coordinate plane. This represents \( 2 \mathbf{v} \).
05
Sketch the Vector -2v
Draw a vector starting from the origin (0,0) to the point (-6, 10) on the same coordinate plane. This represents \( -2 \mathbf{v} \). Explain that the direction of \( -2 \mathbf{v} \) is opposite to \( 2 \mathbf{v} \), while having the same magnitude.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Components
Vector components are crucial in understanding how vectors behave in space. A vector is essentially a quantity that has both magnitude and direction. In the case of two-dimensional vectors, the components help describe the vector's influence on each axis.
The vector \( \mathbf{v} = 3 \mathbf{i} - 5 \mathbf{j} \) includes:
The vector \( \mathbf{v} = 3 \mathbf{i} - 5 \mathbf{j} \) includes:
- The component \( 3 \mathbf{i} \), which represents a movement 3 units parallel to the x-axis.
- The component \( -5 \mathbf{j} \), which signifies moving 5 units in the negative direction, parallel to the y-axis.
Multiplying Vectors
When multiplying vectors by scalar values, you are essentially stretching or shrinking the vector, without changing its direction鈥攗nless the scalar is negative. In our exercise, we calculated \( 2 \mathbf{v} \) and \( -2 \mathbf{v} \).
- For \( 2 \mathbf{v} \): Multiply each component of \( \mathbf{v} \) by 2, resulting in \( 6 \mathbf{i} - 10 \mathbf{j} \). This means the vector is twice as long but keeps the same direction.
- For \( -2 \mathbf{v} \): Multiply each component by -2, creating \( -6 \mathbf{i} + 10 \mathbf{j} \). This vector also becomes double in magnitude but flips in direction.
Sketching Vectors
Sketching vectors is a visual representation of their components and behavior in space. It involves plotting them on a coordinate system based on their components. For the vector \( 2 \mathbf{v} \) and \( -2 \mathbf{v} \):
- \( 2 \mathbf{v} \): Draw a line starting at the origin (0,0) to the point (6, -10). This shows that the vector is directed toward the bottom right quadrant, with a relatively steep incline.
- \( -2 \mathbf{v} \): Begin at the origin and point towards (-6, 10). The vector now directs to the top left quadrant, opposite in direction compared to \( 2 \mathbf{v} \), but with equal magnitude.
Coordinate Plane
The coordinate plane is the backdrop upon which vectors are sketched and visualized. It consists of two axes鈥攖he x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin point (0,0), forming the center reference point.
- The positive direction of the x-axis extends to the right and the positive direction of the y-axis extends upwards.
- The four quadrants on this plane aid in determining vector direction. For example, \( (6, -10) \) resides in the fourth quadrant, while \( (-6, 10) \) lies in the second quadrant.
