30 Questions
What does a vector quantity possess?
Both magnitude and direction
How are vectors represented in this book?
Bold face type
What does |v| represent for a vector v?
Magnitude
How are vectors often represented when written by hand?
With an arrow placed over the letter
When are two vectors A and B considered equal?
When they have the same magnitude and direction
What is represented by the vector PP′?
Displacement vector
What are vectors with fixed locations called?
Local vectors
When multiplying a vector A by a positive number λ, what changes?
Only magnitude
What happens when a vector is displaced parallel to itself?
The vector is unchanged
What are vectors with an important location or line of application called?
Localised vectors
Which law of vector addition do vectors obey by definition?
Triangle law or parallelogram law
What is the dimension of λA when a vector A is multiplied by a scalar λ with physical dimension?
Product of the dimensions of λ and A
In the context of vector addition, what does the angle α represent?
The angle between the two vectors being added
What is the formula for calculating the magnitude of the resultant vector R in terms of vectors A and B?
$R = A^2 + B^2 + 2AB \cos(\theta)$
If PM represents a vector in direction specified by the problem, what does PM equal to?
$A \sin(\alpha) = B \sin(\beta)$
For vector addition using the parallelogram method, what does SN represent in relation to the resultant vector R?
The perpendicular component of the resultant vector R
What type of triangle is used for calculating the magnitude of R using the Law of Cosines?
Scalene triangle
In which direction is the resultant velocity of the boat calculated in the given example?
$60°$ east of south
What is the relationship between velocity and acceleration in one dimension?
They are always in the same direction
For motion in two or three dimensions, what range of angles can exist between velocity and acceleration vectors?
0° to 180°
Given the position of a particle as r = 3.0t i + 2.0t^2 j + 5.0 k, what is the acceleration of the particle at any time t?
4.0 t j
If a particle's velocity at t = 1.0 s is v = 3.0 i + 4.0 j m/s, what is the magnitude of its velocity at that instant?
5.0 m/s
At t = 1.0 s, if a particle's velocity is v = 3.0 i + 4.0 j m/s, what is the direction of the velocity in relation to the x-axis?
+53°
What happens to the average acceleration over an interval of time when the acceleration of an object moving in the x-y plane is constant?
It remains constant
What is the direction θ that R makes with the vertical?
19°
How can a vector be resolved into two component vectors along a set of two vectors?
By using unit vectors
What is the purpose of a unit vector in a rectangular coordinate system?
To specify a direction only
How are unit vectors along the x-, y-, and z-axes denoted?
$î , ĵ , k̂$
What is the magnitude of a unit vector?
1
What happens when a unit vector is multiplied by a scalar?
It scales in magnitude
Study Notes
Scalars and Vectors
- Scalars can be multiplied and divided, but they do not have a fixed location.
- Vectors can be displaced parallel to themselves without changing, making them "free vectors".
- In some physical applications, the location or line of application of a vector is important, making them "localised vectors".
- Multiplying a vector by a scalar with a physical dimension changes the dimension of the resulting vector.
Addition and Subtraction of Vectors
- Vectors can be added and subtracted graphically using the parallelogram method.
- Two vectors are equal if and only if they have the same magnitude and direction.
Multiplication of Vectors by Real Numbers
- Multiplying a vector by a positive number changes its magnitude but not its direction.
- Vectors obey the triangle law or parallelogram law of addition.
Resolving Vectors
- A vector can be resolved into two component vectors along a set of two non-colinear vectors.
- Unit vectors are used to specify directions; they have no dimension or unit.
- Unit vectors along the x-, y-, and z-axes of a rectangular coordinate system are denoted by i, j, and k, respectively.
Vector Quantities
- A vector quantity has both magnitude and direction and obeys the triangle law of addition.
- Examples of vector quantities include displacement, velocity, acceleration, and force.
Displacement Vector
- A displacement vector is a vector that represents the motion of an object from one position to another.
- The magnitude of the displacement vector is the magnitude of the motion.
Resultant Vector
- The resultant vector is the sum of two or more vectors.
- The magnitude and direction of the resultant vector can be calculated using the parallelogram method.
Acceleration
- Acceleration is the rate of change of velocity.
- In one dimension, acceleration is always along the same straight line as velocity.
- In two or three dimensions, velocity and acceleration vectors may have any angle between 0° and 180°.
Test your knowledge on resolving vectors into component vectors and calculating directions in physics. Practice finding angles and components of vectors using given equations and figures.
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