Multiple Choice
Sam walked 4 blocks east and 3 blocks north. How far did Sam walk?
3. Vectors
Introduction to Vectors
Problem 5a
Textbook Question
(II) is a vector 21.8 units in magnitude and points at an angle of 23.4° above the negative axis. Sketch this vector.
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Understand the problem: The vector \( \mathbf{V} \) has a magnitude of 21.8 units and is directed at an angle of 23.4° above the negative \( x \)-axis. This means the vector is in the second quadrant of the Cartesian coordinate system.
Draw the coordinate system: Sketch a standard Cartesian plane with the \( x \)-axis and \( y \)-axis intersecting at the origin. Label the positive and negative directions of both axes.
Locate the angle: Starting from the negative \( x \)-axis, measure an angle of 23.4° counterclockwise. This is the direction of the vector \( \mathbf{V} \).
Draw the vector: From the origin, draw a straight arrow pointing in the direction of the angle you just measured. The length of the arrow should represent the magnitude of the vector (21.8 units). Label the vector as \( \mathbf{V} \).
Indicate the components: Optionally, you can draw dashed lines from the tip of the vector to the \( x \)-axis and \( y \)-axis to represent the vector's components. These components can be calculated as \( V_x = -|V| \cos(23.4°) \) and \( V_y = |V| \sin(23.4°) \), where \( |V| = 21.8 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Representation
A vector is a quantity that has both magnitude and direction. In this case, the vector V has a magnitude of 21.8 units and points at an angle of 23.4° above the negative x-axis. To sketch this vector, one must represent it as an arrow originating from a point, with its length proportional to the magnitude and its direction determined by the specified angle.
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Coordinate System
Understanding the coordinate system is crucial for accurately sketching vectors. The standard Cartesian coordinate system consists of an x-axis (horizontal) and a y-axis (vertical). The negative x-axis indicates the direction opposite to the positive x-axis, and angles are typically measured counterclockwise from the positive x-axis, which is essential for determining the correct orientation of the vector.
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Angle Measurement
Angle measurement is vital in vector analysis, as it defines the direction of the vector in relation to the axes. In this scenario, the angle of 23.4° above the negative x-axis means that the vector is oriented slightly upward from the left side of the graph. This angle can be visualized as a rotation from the negative x-axis towards the positive y-axis, affecting how the vector is drawn in the sketch.
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