Domanda del libro di testo
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3v - 4w
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 4, Problema 31
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3w + 2v
Guida verificata passo dopo passo1
Identify the given vectors: \( \mathbf{u} = 2\mathbf{i} - 5\mathbf{j} \), \( \mathbf{v} = -3\mathbf{i} + 7\mathbf{j} \), and \( \mathbf{w} = -\mathbf{i} - 6\mathbf{j} \).
Calculate the scalar multiplication of vector \( \mathbf{w} \) by 3: multiply each component of \( \mathbf{w} \) by 3, resulting in \( 3\mathbf{w} = 3(-\mathbf{i}) + 3(-6\mathbf{j}) \).
Calculate the scalar multiplication of vector \( \mathbf{v} \) by 2: multiply each component of \( \mathbf{v} \) by 2, resulting in \( 2\mathbf{v} = 2(-3\mathbf{i}) + 2(7\mathbf{j}) \).
Add the resulting vectors from the previous two steps component-wise: add the \( \mathbf{i} \) components together and the \( \mathbf{j} \) components together to find \( 3\mathbf{w} + 2\mathbf{v} \).
Write the final vector in the form \( a\mathbf{i} + b\mathbf{j} \), where \( a \) and \( b \) are the sums of the respective components.
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Vector Representation in Component Form
Vectors in two dimensions can be expressed as a combination of unit vectors i and j, representing the x and y components respectively. For example, u = 2i - 5j means the vector has an x-component of 2 and a y-component of -5. Understanding this form allows for straightforward vector addition and scalar multiplication.
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Percorso guidato
03:55
Position Vectors & Component Form
Scalar Multiplication of Vectors
Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For instance, multiplying vector w by 3 means multiplying both its i and j components by 3, resulting in a new vector scaled in magnitude but with the same direction if the scalar is positive.
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Percorso guidato
05:05Multiplying Vectors By Scalars
Vector Addition
Vector addition is performed by adding corresponding components of two vectors. For example, adding 2v and 3w requires adding the x-components of 2v and 3w together, and similarly for the y-components, resulting in a new vector that combines both directions and magnitudes.
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Percorso guidato
05:29Adding Vectors Geometrically
Pratica correlata
Domanda del libro di testo
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4j
925
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Domanda del libro di testo
In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
||-2v||
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Domanda del libro di testo
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 1.4, b = 2.9, A = 142°
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Domanda del libro di testo
In Exercises 31–32, find the unit vector that has the same direction as the vector v.
v = -i + 2j
856
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Domanda del libro di testo
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 95, c = 125, A = 49°
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