📝 Unit 6: Introduction to Vectors

R² and R³ · Operations · Magnitude · Unit Vectors
🔄 Not Graded
Purpose: Self-check on geometric and Cartesian vector operations.
Score: 0 / 12
Topic 6.1 — Magnitude & Direction
Question 1
Find \(|\vec u|\) where \(\vec u=(3,-4)\).
Solution:
\(|\vec u|=\sqrt{9+16}=5\).
Question 2
Magnitude of \(\vec v=(1,2,2)\).
Solution:
\(\sqrt{1+4+4}=3\).
Question 3
Direction angle (from positive x-axis) of \(\vec u=(\sqrt 3,1)\), in degrees.
Solution:
\(\tan\theta=1/\sqrt 3\) → \(\theta=30°\).
Topic 6.2 — Vector Operations
Question 4
If \(\vec a=(2,-1,3)\) and \(\vec b=(0,4,-2)\), find the second component of \(2\vec a+\vec b\).
Solution:
\(2\vec a=(4,-2,6)\); \(+\vec b=(4,2,4)\). Second component = 2.
Question 5
Find \(\vec a-\vec b\) for \(\vec a=(5,2),\,\vec b=(3,-1)\). Magnitude:
Solution:
\((2,3)\); \(\sqrt{4+9}=\sqrt{13}\approx 3.606\).
Topic 6.3 — Unit Vectors
Question 6
Find the first component of the unit vector in the direction of \((6,8)\).
Solution:
\(|\vec v|=10\); \(\hat v=(0.6,0.8)\).
Question 7
A vector with magnitude 15 in direction of \((1,-2,2)\). Its first component is:
Solution:
\(|\vec v|=3\); \(\hat v=(1/3,-2/3,2/3)\); \(15\hat v=(5,-10,10)\).
Topic 6.4 — Geometric Vectors
Question 8
Two forces of magnitudes 6 N and 8 N act perpendicular to each other. The magnitude of the resultant is:
Solution:
Pythagoras: \(\sqrt{36+64}=10\) N.
Question 9
If \(|\vec a|=5\), \(|\vec b|=4\), and the angle between them is \(60°\), the magnitude of \(\vec a+\vec b\) is:
Solution:
\(|\vec a+\vec b|^2=|\vec a|^2+|\vec b|^2+2|\vec a||\vec b|\cos\theta=25+16+40(0.5)=61\). \(\sqrt{61}\approx 7.81\).
Question 10
Two vectors are equal if and only if:
Solution:
Vectors are determined by magnitude and direction (location is irrelevant).
Topic 6.5 — Position Vectors
Question 11
Vector from \(A(2,-1,3)\) to \(B(5,1,7)\). Magnitude:
Solution:
\(\vec{AB}=(3,2,4)\); \(\sqrt{9+4+16}=\sqrt{29}\approx 5.39\).
Question 12
For \(\vec u=(1,2)\) and \(\vec v=(3,1)\), are they parallel?
Solution:
\(1/3 \ne 2/1\) → not scalar multiples → not parallel.

📊 Self-Reflection

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