Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4i
1
views
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Not the one you use?Change textbookSelect a different textbook
Table of contents
Chapter 4, Problem 30
In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
||-2v||
Verified step by step guidance1
Identify the given vector \( \mathbf{v} = \mathbf{i} - 5\mathbf{j} \), which can be written in component form as \( \mathbf{v} = (1, -5) \).
Calculate the scalar multiplication of the vector \( \mathbf{v} \) by \( -2 \), which gives \( -2\mathbf{v} = -2(1, -5) = (-2, 10) \).
Recall that the magnitude (or norm) of a vector \( \mathbf{a} = (x, y) \) is given by the formula:
\[ \\| \mathbf{a} \\| = \sqrt{x^2 + y^2} \]
Apply the magnitude formula to the vector \( -2\mathbf{v} = (-2, 10) \) by substituting \( x = -2 \) and \( y = 10 \) into the formula:
\[ \\| -2\mathbf{v} \\| = \sqrt{(-2)^2 + 10^2} \]
Simplify the expression under the square root to find the magnitude of \( -2\mathbf{v} \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Notation and Components
Vectors are represented using unit vectors i and j to denote their components along the x and y axes, respectively. For example, v = i - 5j means the vector has an x-component of 1 and a y-component of -5. Understanding this notation is essential for performing vector operations.
Recommended video:
06:01
i & j Notation
Scalar Multiplication of Vectors
Scalar multiplication involves multiplying each component of a vector by a scalar value. For instance, multiplying vector v by -2 results in a new vector with components scaled by -2, changing both magnitude and direction accordingly.
Recommended video:
05:05Multiplying Vectors By Scalars
Magnitude (Norm) of a Vector
The magnitude of a vector is the length or size of the vector, calculated using the Pythagorean theorem as the square root of the sum of the squares of its components. For vector v = (x, y), ||v|| = √(x² + y²). This concept is crucial for finding ||-2v||.
Recommended video:
04:44Finding Magnitude of a Vector
Related Practice
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
-4w
1
views
Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4j
Textbook Question
In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.
a = 11 yards, b = 9 yards, c = 7 yards
Textbook Question
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°
1
views
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3w + 2v
1
views