Textbook Question
In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i - 10j
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 46
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = i - j
Verified step by step guidance1
Identify the given vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \), which can be written in component form as \( \mathbf{v} = \langle 1, -1 \rangle \).
Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula \( \| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2} \). Substitute the components to get \( \| \mathbf{v} \| = \sqrt{1^2 + (-1)^2} \).
Simplify the expression under the square root to find the magnitude \( \| \mathbf{v} \| \).
Find the unit vector \( \mathbf{u} \) in the same direction as \( \mathbf{v} \) by dividing each component of \( \mathbf{v} \) by its magnitude: \( \mathbf{u} = \frac{1}{\| \mathbf{v} \|} \mathbf{v} = \left\langle \frac{1}{\| \mathbf{v} \|}, \frac{-1}{\| \mathbf{v} \|} \right\rangle \).
Express the unit vector \( \mathbf{u} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{u} = \frac{1}{\| \mathbf{v} \|} \mathbf{i} - \frac{1}{\| \mathbf{v} \|} \mathbf{j} \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 is essential to find a unit vector that aligns with the original vector.
Recommended video:
05:13
Finding Direction of a Vector
Vector Magnitude (Length)
The magnitude of a vector is its length, calculated using the square root of the sum of the squares of its components. For vector v = i - j, the magnitude is √(1² + (-1)²) = √2. This value is used to normalize the vector.
Recommended video:
04:44Finding Magnitude of a Vector
Unit Vector
A unit vector has a magnitude of 1 and points in a specific direction. To find a unit vector in the same direction as v, divide each component of v by its magnitude. This process scales the vector to length one without changing its direction.
Recommended video:
04:04Unit Vector in the Direction of a Given Vector
Related Practice
Textbook Question
In Exercises 45–46, find the area of the triangle with the given vertices. Round to the nearest square unit. (-2, -3), (-2, 2), (2, 1)
2
views
Textbook Question
Write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given.
||v|| = 8, θ = 45°
Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = i + j
Textbook Question
In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 6, θ = 30°
Textbook Question
In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i + 10j