Domanda del libro di testo
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. 3v - 4w
646
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 6
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. v ⋅ w
Guida verificata passo dopo passo1
Recall that the dot product of two vectors \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \) and \( \mathbf{w} = w_1 \mathbf{i} + w_2 \mathbf{j} \) is given by the formula:
\[ \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \]
Identify the components of the given vectors:
For \( \mathbf{v} = -5 \mathbf{i} + 2 \mathbf{j} \), we have \( v_1 = -5 \) and \( v_2 = 2 \).
For \( \mathbf{w} = 2 \mathbf{i} - 4 \mathbf{j} \), we have \( w_1 = 2 \) and \( w_2 = -4 \).
Substitute the components into the dot product formula:
\[ \mathbf{v} \cdot \mathbf{w} = (-5)(2) + (2)(-4) \]
Simplify the expression by performing the multiplications and then adding the results:
\[ (-5)(2) + (2)(-4) = -10 + (-8) \]
Add the two products to find the dot product value:
\[ -10 + (-8) = -18 \]
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:
1mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Dot Product of Vectors
The dot product is an algebraic operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components and summing the results, e.g., for vectors v = (v1, v2) and w = (w1, w2), v ⋅ w = v1w1 + v2w2.
Video consigliato:
Percorso guidato
05:40
Introduction to Dot Product
Vector Components and Notation
Vectors in two dimensions are expressed in terms of unit vectors i and j, representing the x and y directions respectively. Understanding how to identify and manipulate these components is essential for performing operations like the dot product.
Video consigliato:
Percorso guidato
06:01i & j Notation
Geometric Interpretation of the Dot Product
The dot product relates to the angle between two vectors: v ⋅ w = |v||w|cosθ, where θ is the angle between v and w. This relationship helps in finding the angle or understanding the projection of one vector onto another.
Video consigliato:
Percorso guidato
05:40Introduction to Dot Product
Pratica correlata
Domanda del libro di testo
In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. C = 50°, a = 3, c = 1
722
views
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.
471
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.
c. Find the exact value of the expression.
cos 50° cos 20° + sin 50° sin 20°
817
views
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.
470
views
Domanda del libro di testo
In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = i - j
803
views