Textbook Question
A plane has an airspeed of 520 mph. The pilot wishes to fly on a bearing of 310°. A wind of 37 mph is blowing from a bearing of 212°. In what direction should the pilot fly, and what will be her ground speed?
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8. Vectors
Geometric Vectors
2:26 minutesProblem 24
Textbook Question
In Exercises 22–24, sketch each vector as a position vector and find its magnitude.
v = -3j
Verified step by step guidance1
Identify the vector given: \( \mathbf{v} = -3\mathbf{j} \). This means the vector has no \( \mathbf{i} \) (x-direction) component and a \( -3 \) component in the \( \mathbf{j} \) (y-direction).
Express the vector in component form as \( \mathbf{v} = (0, -3) \), where 0 is the x-component and -3 is the y-component.
To sketch the vector as a position vector, start at the origin \( (0,0) \) and draw an arrow pointing straight down to the point \( (0, -3) \) on the Cartesian plane.
Recall that the magnitude (or length) of a vector \( \mathbf{v} = (x, y) \) is given by the formula \( \| \mathbf{v} \| = \sqrt{x^2 + y^2} \).
Calculate the magnitude of \( \mathbf{v} \) by substituting the components: \( \| \mathbf{v} \| = \sqrt{0^2 + (-3)^2} = \sqrt{9} \). This gives the magnitude of the vector.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Position Vector
A position vector represents the location of a point in space relative to the origin. It is typically expressed in component form, such as v = ai + bj, where i and j are unit vectors along the x and y axes. In this question, the vector v = -3j points 3 units in the negative y-direction.
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Position Vectors & Component Form
Vector Components and Unit Vectors
Vectors are broken down into components along coordinate axes using unit vectors i (x-axis) and j (y-axis). The given vector v = -3j has no x-component and a y-component of -3, indicating direction and magnitude along the y-axis only.
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Magnitude of a Vector
The magnitude of a vector is its length and is found using the Pythagorean theorem: |v| = √(a² + b²). For v = -3j, the magnitude is |v| = √(0² + (-3)²) = 3, representing the distance from the origin to the point defined by the vector.
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