Domanda del libro di testo
In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i - 10j
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 4, Problema 44
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 4i - 2j
Guida verificata passo dopo passo1
Identify the given vector \( \mathbf{v} = 4\mathbf{i} - 2\mathbf{j} \). This means the vector has components \( (4, -2) \).
Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula:
\[ \|\mathbf{v}\| = \sqrt{(4)^2 + (-2)^2} \]
Simplify the expression under the square root to find the magnitude:
\[ \|\mathbf{v}\| = \sqrt{16 + 4} \]
Find the unit vector \( \mathbf{u} \) in the same direction as \( \mathbf{v} \) by dividing each component of \( \mathbf{v} \) by its magnitude:
\[ \mathbf{u} = \left( \frac{4}{\|\mathbf{v}\|}, \frac{-2}{\|\mathbf{v}\|} \right) \]
Express the unit vector in terms of the standard unit vectors \( \mathbf{i} \) and \( \mathbf{j} \):
\[ \mathbf{u} = \frac{4}{\|\mathbf{v}\|} \mathbf{i} - \frac{2}{\|\mathbf{v}\|} \mathbf{j} \]
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Vector Components
A vector in two dimensions can be expressed in terms of its components along the x and y axes, typically written as v = ai + bj, where a and b are scalar values. Understanding these components is essential for operations like finding magnitude and direction.
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Percorso guidato
03:55
Position Vectors & Component Form
Magnitude of a Vector
The magnitude (or length) of a vector v = ai + bj is calculated using the Pythagorean theorem as √(a² + b²). This scalar value represents the distance from the origin to the point defined by the vector components.
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Percorso guidato
04:44Finding Magnitude of a Vector
Unit Vector
A unit vector has a magnitude of 1 and points in the same direction as the original vector. It is found by dividing each component of the vector by its magnitude, effectively normalizing the vector without changing its direction.
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Percorso guidato
04:04Unit Vector in the Direction of a Given Vector
Pratica correlata
Domanda del libro di testo
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 3i - 2j
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Domanda del libro di testo
In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = i + j
989
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Domanda del libro di testo
In Exercises 43–44, find the angle, in degrees, between v and w.
v = 2 cos(4π/3) i + 2 sin(4π/3) j, w = 3 cos(3π/2) i + 3 sin(3π/2) j
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Domanda del libro di testo
In Exercises 42–43, find projᵥᵥv. Then decompose v into two vectors, v₁ and v₂ where v₁ is parallel to w and v₂ is orthogonal to w.
v = -2i + 5j, w = 5i + 4j
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Domanda del libro di testo
In Exercises 43–44, use the given measurements to solve the following triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree. a = 400, b = 300
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