Domanda del libro di testo
In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 12, θ = 225°
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 4, Problema 47
In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i + 10j
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Identify the vectors \( \mathbf{v} = 3\mathbf{i} - 5\mathbf{j} \) and \( \mathbf{w} = 6\mathbf{i} + 10\mathbf{j} \). Write them in component form as \( \mathbf{v} = (3, -5) \) and \( \mathbf{w} = (6, 10) \).
To check if the vectors are parallel, see if one is a scalar multiple of the other. This means checking if there exists a scalar \( k \) such that \( (6, 10) = k(3, -5) \).
To check if the vectors are orthogonal (perpendicular), calculate their dot product. The dot product formula is \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \). Substitute the components to get \( 3 \times 6 + (-5) \times 10 \).
Evaluate the dot product expression to determine if it equals zero. If the dot product is zero, the vectors are orthogonal.
Based on the results from the scalar multiple check and the dot product, conclude whether \( \mathbf{v} \) and \( \mathbf{w} \) are parallel, orthogonal, or neither.
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Vector Representation in Component Form
Vectors can be expressed in terms of their components along the coordinate axes, such as v = 3i - 5j, where 3 and -5 are the components along the x and y axes respectively. Understanding this form allows for algebraic operations like addition, scalar multiplication, and dot product calculation.
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Percorso guidato
03:55
Position Vectors & Component Form
Parallel Vectors
Two vectors are parallel if one is a scalar multiple of the other, meaning their components are proportional. For example, if v = k * w for some scalar k, then v and w point in the same or opposite direction, indicating parallelism.
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Percorso guidato
03:48Introduction to Vectors
Orthogonal Vectors and the Dot Product
Vectors are orthogonal (perpendicular) if their dot product equals zero. The dot product is calculated by multiplying corresponding components and summing the results. If v · w = 0, the vectors form a 90-degree angle.
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05:40Introduction to Dot Product
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Domanda del libro di testo
In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i + 18 j 5
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Domanda del libro di testo
Write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given.
||v|| = 8, θ = 45°
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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
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Domanda del libro di testo
In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 6, θ = 30°
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