**Vectors Problem Explanation** --- **Given:** - \( u = 1i - 3j \) - \( v = 2i + 3j \) **Tasks:** a) Calculate \( 2(u) - 3(v) \) \[ 2(1i - 3j) - 3(2i + 3j) \] \[ = (2i - 6j) - (6i + 9j) \] \[ = 2i - 6j - 6i - 9j \] \[ = -4i - 15j \] b) Calculate \( (i - 3j) \cdot (2i + 3j) \) - This involves finding the dot product: The dot product is calculated as: \[ (1 \cdot 2) + (-3 \cdot 3) = 2 - 9 = -7 \] c) Find a unit vector in the same direction as \( v \) - To find a unit vector, divide each component of \( v \) by its magnitude. - First, calculate the magnitude of \( v \): \[ \|v\| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \] - The unit vector in the direction of \( v \) is: \[ \frac{1}{\sqrt{13}}(2i + 3j) = \left(\frac{2}{\sqrt{13}}i + \frac{3}{\sqrt{13}}j\right) \] --- This set of problems helps understand vector arithmetic and operations such as scaling, subtraction, and dot product, as well as finding unit vectors.
**Vectors Problem Explanation** --- **Given:** - \( u = 1i - 3j \) - \( v = 2i + 3j \) **Tasks:** a) Calculate \( 2(u) - 3(v) \) \[ 2(1i - 3j) - 3(2i + 3j) \] \[ = (2i - 6j) - (6i + 9j) \] \[ = 2i - 6j - 6i - 9j \] \[ = -4i - 15j \] b) Calculate \( (i - 3j) \cdot (2i + 3j) \) - This involves finding the dot product: The dot product is calculated as: \[ (1 \cdot 2) + (-3 \cdot 3) = 2 - 9 = -7 \] c) Find a unit vector in the same direction as \( v \) - To find a unit vector, divide each component of \( v \) by its magnitude. - First, calculate the magnitude of \( v \): \[ \|v\| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \] - The unit vector in the direction of \( v \) is: \[ \frac{1}{\sqrt{13}}(2i + 3j) = \left(\frac{2}{\sqrt{13}}i + \frac{3}{\sqrt{13}}j\right) \] --- This set of problems helps understand vector arithmetic and operations such as scaling, subtraction, and dot product, as well as finding unit vectors.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
Hi I just need help with b & c
![**Vectors Problem Explanation**
---
**Given:**
- \( u = 1i - 3j \)
- \( v = 2i + 3j \)
**Tasks:**
a) Calculate \( 2(u) - 3(v) \)
\[
2(1i - 3j) - 3(2i + 3j)
\]
\[
= (2i - 6j) - (6i + 9j)
\]
\[
= 2i - 6j - 6i - 9j
\]
\[
= -4i - 15j
\]
b) Calculate \( (i - 3j) \cdot (2i + 3j) \)
- This involves finding the dot product:
The dot product is calculated as:
\[
(1 \cdot 2) + (-3 \cdot 3) = 2 - 9 = -7
\]
c) Find a unit vector in the same direction as \( v \)
- To find a unit vector, divide each component of \( v \) by its magnitude.
- First, calculate the magnitude of \( v \):
\[
\|v\| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\]
- The unit vector in the direction of \( v \) is:
\[
\frac{1}{\sqrt{13}}(2i + 3j) = \left(\frac{2}{\sqrt{13}}i + \frac{3}{\sqrt{13}}j\right)
\]
---
This set of problems helps understand vector arithmetic and operations such as scaling, subtraction, and dot product, as well as finding unit vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fe22a46-7618-4b05-a024-d92b45c5cc20%2F6af1f35c-8f13-459f-909b-c458e68205f6%2Fhdnafi.jpeg&w=3840&q=75)
Transcribed Image Text:**Vectors Problem Explanation**
---
**Given:**
- \( u = 1i - 3j \)
- \( v = 2i + 3j \)
**Tasks:**
a) Calculate \( 2(u) - 3(v) \)
\[
2(1i - 3j) - 3(2i + 3j)
\]
\[
= (2i - 6j) - (6i + 9j)
\]
\[
= 2i - 6j - 6i - 9j
\]
\[
= -4i - 15j
\]
b) Calculate \( (i - 3j) \cdot (2i + 3j) \)
- This involves finding the dot product:
The dot product is calculated as:
\[
(1 \cdot 2) + (-3 \cdot 3) = 2 - 9 = -7
\]
c) Find a unit vector in the same direction as \( v \)
- To find a unit vector, divide each component of \( v \) by its magnitude.
- First, calculate the magnitude of \( v \):
\[
\|v\| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\]
- The unit vector in the direction of \( v \) is:
\[
\frac{1}{\sqrt{13}}(2i + 3j) = \left(\frac{2}{\sqrt{13}}i + \frac{3}{\sqrt{13}}j\right)
\]
---
This set of problems helps understand vector arithmetic and operations such as scaling, subtraction, and dot product, as well as finding unit vectors.
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