Domanda del libro di testo
In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
456
views
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
Capitolo 4, Problema 4.40
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = -5j
Guida verificata passo dopo passo1
Identify the given vector \( \mathbf{v} = -5\mathbf{j} \). This means the vector points in the negative y-direction with a magnitude of 5.
Recall that the unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} \) is found by dividing \( \mathbf{v} \) by its magnitude \( \|\mathbf{v}\| \). The formula is:
\[ \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} \]
Calculate the magnitude of \( \mathbf{v} \). Since \( \mathbf{v} = 0\mathbf{i} - 5\mathbf{j} \), the magnitude is:
\[ \|\mathbf{v}\| = \sqrt{0^2 + (-5)^2} = \sqrt{25} \]
Divide each component of \( \mathbf{v} \) by the magnitude \( \|\mathbf{v}\| \) to get the unit vector components:
\[ \mathbf{u} = \left( \frac{0}{\|\mathbf{v}\|}, \frac{-5}{\|\mathbf{v}\|} \right) \]
Write the unit vector \( \mathbf{u} \) in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) using the components found in the previous step.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Vector Direction
The direction of a vector is the orientation in space it points to, regardless of its magnitude. Two vectors have the same direction if one is a scalar multiple of the other. Understanding direction helps in finding unit vectors that preserve this orientation.
Video consigliato:
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 for components. Magnitude is essential for normalizing a vector to find its unit vector.
Video consigliato:
Percorso guidato
04:44Finding Magnitude of a Vector
Unit Vector
A unit vector has a magnitude of exactly one and points in a specific direction. To find a unit vector in the same direction as a given vector, divide the vector by its magnitude. This process normalizes the vector, preserving direction but standardizing length.
Video consigliato:
Percorso guidato
04:04Unit Vector in the Direction of a Given Vector
Pratica correlata
Domanda del libro di testo
Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.
P₁ = (-1, 6), P₂ = (7, -5)
446
views
Domanda del libro di testo
In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β.
b. Write the expression as the cosine of an angle.
cos 50° cos 20° + sin 50° sin 20°
790
views
Domanda del libro di testo
The magnitude and direction angle of v are ||v|| = 12 and θ = 60°. Express v in terms of i and j.
461
views
Domanda del libro di testo
If P₁ = (-2, 3), P₂ = (-1, 5), and v is the vector from P₁ to P₂, Write v in terms of i and j.
869
views
Domanda del libro di testo
In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.
864
views