Ch. 7 - Applications of Trigonometry and Vectors

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 7 - Applications of Trigonometry and Vectors

Problema 14

Capitolo 8, Problema 14

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary. See Example 1.
〈8√2, -8√2〉

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Identify the components of the vector. Here, the vector is given as \(\langle 8\sqrt{2}, -8\sqrt{2} \rangle\), where the x-component is \(8\sqrt{2}\) and the y-component is \(-8\sqrt{2}\).

Calculate the magnitude of the vector using the formula \(\text{magnitude} = \sqrt{x^2 + y^2}\). Substitute the components: \(\sqrt{(8\sqrt{2})^2 + (-8\sqrt{2})^2}\).

Simplify the expression inside the square root by squaring each component and adding them together.

Find the direction angle \(\theta\) using the formula \(\theta = \tan^{-1}\left( \frac{y}{x} \right)\). Substitute the components: \(\theta = \tan^{-1}\left( \frac{-8\sqrt{2}}{8\sqrt{2}} \right)\).

Determine the correct quadrant for the angle based on the signs of the x and y components, then adjust the angle accordingly to find the direction angle measured counterclockwise from the positive x-axis. Round the angle to the nearest tenth of a degree.

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Vector Magnitude

The magnitude of a vector represents its length and is calculated using the Pythagorean theorem. For a vector with components (x, y), the magnitude is √(x² + y²). This gives a non-negative scalar value indicating the vector's size regardless of direction.

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Percorso guidato

04:44

Finding Magnitude of a Vector

Direction Angle of a Vector

The direction angle of a vector is the angle it makes with the positive x-axis, measured counterclockwise. It can be found using the inverse tangent function: θ = arctan(y/x). Adjustments may be needed based on the vector's quadrant to get the correct angle.

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Percorso guidato

05:13

Finding Direction of a Vector

Handling Vectors with Radical Components

Vectors with components involving radicals, like √2, require careful arithmetic when calculating magnitude and direction. Simplifying expressions and using exact values before rounding helps maintain accuracy in the final results.

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Percorso guidato

03:55

Position Vectors & Component Form

Pratica correlata

Domanda del libro di testo

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈-4, 4√3〉

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Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.

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a + (b + c)

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c + d

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Domanda del libro di testo

Solve each triangle. Approximate values to the nearest tenth.

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