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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TOC.tableOfContents
Capitolo 4, Problema 32
In Exercises 31–32, find the unit vector that has the same direction as the vector v.
v = -i + 2j
Guida verificata passo dopo passo1
Identify the given vector \( \mathbf{v} = -\mathbf{i} + 2\mathbf{j} \), which can be written in component form as \( \mathbf{v} = (-1, 2) \).
Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula:
\[ \text{magnitude} = ||\mathbf{v}|| = \sqrt{(-1)^2 + 2^2} \]
Simplify the expression under the square root to find the magnitude:
\[ ||\mathbf{v}|| = \sqrt{1 + 4} \]
Find the unit vector \( \mathbf{u} \) by dividing each component of \( \mathbf{v} \) by its magnitude:
\[ \mathbf{u} = \left( \frac{-1}{||\mathbf{v}||}, \frac{2}{||\mathbf{v}||} \right) \]
Express the unit vector in terms of \( \mathbf{i} \) and \( \mathbf{j} \) as:
\[ \mathbf{u} = \frac{-1}{||\mathbf{v}||} \mathbf{i} + \frac{2}{||\mathbf{v}||} \mathbf{j} \]
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Vector Direction
The direction of a vector is the orientation it points to in space, independent of its length. Two vectors have the same direction if one is a scalar multiple of the other. Understanding direction helps in finding a unit vector that points the same way as the original vector.
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Percorso guidato
05:13
Finding Direction of a Vector
Magnitude of a Vector
The magnitude (or length) of a vector is the distance from the origin to the point represented by the vector. It is calculated using the Pythagorean theorem as the square root of the sum of the squares of its components. Magnitude is essential for normalizing a vector to unit length.
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Percorso guidato
04:44Finding Magnitude of a Vector
Unit Vector
A unit vector has a magnitude of exactly one and indicates direction only. To find a unit vector in the same direction as a given vector, divide each component of the vector by its magnitude. This process is called normalization and preserves direction while standardizing length.
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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 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 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 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.
A = 22°, b = 20 feet, c = 50 feet
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Domanda del libro di testo
In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w. v = 3i - 2j, w = i - j
769
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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.
3w + 2v
825
views