Ch. 4 - Laws of Sines and Cosines; Vectors

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 4 - Laws of Sines and Cosines; Vectors

Problem 4.33

Chapter 4, Problem 4.33

The magnitude and direction angle of v are ||v|| = 12 and θ = 60°. Express v in terms of i and j.

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Recall that a vector \( \mathbf{v} \) in two dimensions can be expressed in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \), where \( v_x \) and \( v_y \) are the components of \( \mathbf{v} \) along the x-axis and y-axis respectively.

Use the magnitude \( ||\mathbf{v}|| = 12 \) and the direction angle \( \theta = 60^\circ \) to find the components. The x-component is given by \( v_x = ||\mathbf{v}|| \cos \theta \).

Similarly, the y-component is given by \( v_y = ||\mathbf{v}|| \sin \theta \).

Substitute the known values into the component formulas: \( v_x = 12 \cos 60^\circ \) and \( v_y = 12 \sin 60^\circ \).

Write the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} \) using the components found.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Magnitude and Direction

A vector's magnitude represents its length or size, while the direction angle indicates the angle it makes with the positive x-axis. Together, these define the vector's position in the plane, allowing conversion between polar and rectangular forms.

Unit Vectors i and j

Unit vectors i and j represent the standard basis vectors along the x-axis and y-axis, respectively. Expressing a vector in terms of i and j means breaking it down into horizontal and vertical components.

Conversion from Polar to Rectangular Coordinates

To express a vector given by magnitude and angle in terms of i and j, use trigonometric functions: the x-component is magnitude times cosine of the angle, and the y-component is magnitude times sine of the angle.

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Related Practice

Textbook Question

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

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Textbook Question

Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.

P₁ = (2, -5), P₂ = (-6, 6)

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Textbook Question

Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.

P₁ = (-1, 6), P₂ = (7, -5)

Textbook Question

If P₁ = (-2, 3), P₂ = (-1, 5), and v is the vector from P₁ to P₂, Write v in terms of i and j.

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Textbook Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = -5j

Textbook Question

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.