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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblema 46
Capitolo 8, Problema 46

Write each vector in the form a i + b j.
〈6, -3〉

Guida verificata passo dopo passo
1
Understand that the vector given in angle bracket notation 〈x, y〉 can be expressed in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), where \( \mathbf{i} \) is the unit vector in the x-direction and \( \mathbf{j} \) is the unit vector in the y-direction.
Identify the components of the vector: here, the x-component is 6 and the y-component is -3.
Write the vector as a linear combination of the unit vectors: multiply the x-component by \( \mathbf{i} \) and the y-component by \( \mathbf{j} \).
Express the vector as \( 6\mathbf{i} + (-3)\mathbf{j} \).
Simplify the expression by writing it as \( 6\mathbf{i} - 3\mathbf{j} \), which is the vector in the form \( a\mathbf{i} + b\mathbf{j} \).

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Vector Representation in Component Form

A vector in two dimensions can be expressed as a combination of its horizontal and vertical components. These components correspond to the vector's projections along the x-axis and y-axis, respectively, and are typically written as a multiple of unit vectors i (x-direction) and j (y-direction).
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Position Vectors & Component Form

Unit Vectors i and j

Unit vectors i and j are standard basis vectors in the plane, where i represents a vector of length one in the positive x-direction, and j represents a vector of length one in the positive y-direction. Any 2D vector can be written as a linear combination of i and j.
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i & j Notation

Converting Coordinate Notation to Vector Form

Given a vector in coordinate form 〈x, y〉, it can be rewritten as x i + y j by multiplying the x-component by i and the y-component by j. This form clearly shows the vector's direction and magnitude along each axis.
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Position Vectors & Component Form
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