1.
Find the magnitude and direction of a vector whose initial and terminal points are given by the coordinates (6cm, 1.5cm) and (2.5cm, 5cm).
Exercises 2.3 Exercise
2.
If the velocity vector is \(\vec{v}= (4.000 \hat{i} + 3.000 \hat{j} + 0.100 \hat{k})\) km/h, what is the unit vector of its direction of motion?
3.
A stone is pulled by three strings with forces \(\vec{F}_{1}=\left(10\hat{i}-5\hat{j}+2\hat{k}\right)\) N; \(\vec{F}_{2}=\left(-15\hat{i}-2\hat{k}\right)\) N; and \(\vec{F}_{3}=\left(-5\hat{i}-5\hat{j}\right)\) N. Find the angle between \(\vec{F}_{1}\) and \(\vec{F}_{2}\text{.}\)
4.
Show that when \(\vec{A} + \vec{B} = \vec{C}\text{,}\) then \(C = A^{2} + B^{2} + 2AB \cos\phi \text{,}\) where \(\phi\) is the angle between vectors \(\vec{A}\) and \(\vec{B}\text{.}\)
5.
Vectors \(\vec{A}\) and \(\vec{B}\) are two orthogonal vectors in the xy-plane and they have identical magnitudes. If \(\vec{A} = 3.0 \hat{i} + 4.0 \hat{j}\text{,}\) find \(\vec{B}\text{.}\)
6.
What is the components of a vector \(\vec{a} = (3.0 \hat{i} + 4.0 \hat{j} + 10.0 \hat{k})\) along the vector \(\vec{b} = (1.0 \hat{i} + 4.0 \hat{j})\text{?}\)
7.
If \(\vec{a} = (2 \hat{i} - 4 \hat{j} + \hat{k})\) and \(\vec{b} = (3 \hat{i} + 4 \hat{j} + 10 \hat{k})\text{.}\) Find
- \(\vec{a}\times\vec{b} \) and
- \(\displaystyle \vec{a}\cdot\vec{b}\)
8.
Given the three vectors \(\vec{A} = 4 \hat{i} - 6 \hat{j}\text{,}\) \(\vec{B} = -5 \hat{i} + 5 \hat{j}\text{,}\) \(\vec{C} = 12 \hat{i} + 5 \hat{j} \text{.}\)
9.
- Find the vector \(\vec{D} = \vec{A}+\vec{B}+\vec{C}\text{.}\)
- Find the vector \(\vec{E} = \vec{A} - \vec{B} -\vec{C}\text{.}\)
- Find the scalars \(a\) and \(b\) such that \(a \vec{A} + b \vec{B} + \vec{C}= 0\text{.}\)
10.
Given two points whose position vectors in polar coordinates are \(r_{1} = 20 m\text{,}\) \(\theta_{1} = 120^{o}\) and \(r_{2} = 15 m\text{,}\) \(\theta_{2} = 270^{o}\text{,}\) respectively. Find
- \(r_{1}\text{;}\)
- \(r_{2}\) and
- the distance between the two points.
11.
A car moves 100 m \(60^{o}\) north of east, then it moves 300 m \(30^{o}\) south of west (i.e., \(210^{o}\) from the x-axis). Find the total displacement of the car.
12.
Suppose you walk from point P to point Q. The curved part of your path is a semicircle. What is the magnitude of your displacement from point P to point Q?
13.
Given the forces: \(\vec{F}_{1} = 5 \,N \) at \(30^{o}\) N of E, \(\vec{F}_{2} = 7 \,N \) at \(60^{o}\) S of W, and \(\vec{F}_{3} = 10 \,N \) at \(40^{o} \) E of S.
- Find the components of each force,
- Find the vector components of resultant force, and
- Find the magnitude and direction of resultant force.
14.
If \(\vec{A}= 30(-\hat{k})\) and \(\vec{B}= 60(\hat{i})\text{.}\) Find
- \(\displaystyle \vec{A}\cdot\vec{B}\)
- \(\vec{A}\times\vec{B}\text{.}\)
