34 Questions
What is the direction angle of a vector 𝐴 = 10.0 N, 30.0° N of E?
60.0° E of N
When determining the correct direction angle, what should you start with?
A vector diagram
What is the direction of the vector 𝐴 = 10.0 N, 60.0° E of N?
North of East
How do you measure the direction angle in the first quadrant?
Counterclockwise from the +x-axis
What is the direction of the vector 𝐴 = 10.0 N, 30.0° W of N?
West of North
How do you measure the direction angle in the second quadrant?
Clockwise from the +x-axis
What is the direction of the vector 𝐴 = 10.0 N, 30.0° E of S?
East of South
What is the x-component of vector A in terms of its magnitude and angle θ?
Acos θ
What is the significance of examining the signs of the components of the vector?
To determine the quadrant of the vector
In which quadrant would a vector have a positive x-component and a negative y-component?
IV
What is the y-component of vector A in terms of its magnitude and angle ϕ?
Asin ϕ
If the x-component of a vector is negative, in which quadrant would it lie?
II or III
What is the relationship between the magnitude of a vector and its components?
The magnitude is the hypotenuse of a right triangle with its components
What is the direction of the angle measured from the +x-axis?
Clockwise
What is the y-component of vector A in terms of its magnitude and angle θ?
Asin θ
What is the value of the x-component of vector A?
3.75m
What is the magnitude of vector A?
7.1m
What is the purpose of the unit vector of a vector?
To indicate the direction of the vector
What is the magnitude of the unit vector?
1
What is the result of dividing the perpendicular component of a vector by its magnitude?
+1 or -1
What is the formula for the unit vector?
a@ = A/A
What is the unit vector for the x-coordinate?
ı̂
What is the relationship between the magnitude of a vector and its unit vector?
The unit vector has a magnitude of 1, and the magnitude of a vector is a scalar value
Which of the following is NOT a correct way to write the component vector of A in the x-direction?
A $ ȷ̂
What is the full decomposition of the vector A?
A = A $ ı̂ + A % ȷ̂
In a 3D rectangular coordinate system, which unit vector is along the x-axis?
ı̂
What is the purpose of unit vectors in decomposing a vector?
To provide a useful way to write component vectors
Which of the following is a correct way to write the y-component of a vector?
A % ȷ̂
What is a characteristic of a scalar quantity?
It can be completely described by a magnitude alone.
What is an example of a vector quantity?
Displacement, 20 m from the origin
What is the equation for the magnitude of a vector 𝐴⃗?
𝐴 = |𝐴⃗|
How is a vector quantity represented in analytic form?
𝐴⃗ = 𝐴𝑎
What determines the component form of a vector representation?
The coordinate system used.
What is the symbol used to represent a vector quantity?
𝐴⃗
Study Notes
Scalar and Vector
- Scalars are quantities that can be described by a magnitude (or number and units) alone, such as speed, mass, and temperature.
- Vectors are quantities that can be completely described by both a magnitude and a direction in space, such as displacement, velocity, and force.
Vector Representations
- Vectors can be represented in magnitude-direction form: A⃗ = Aa,whereAisthemagnitudeanda, where A is the magnitude and a,whereAisthemagnitudeanda is the direction.
- Vectors can also be represented in component form: A⃗ = Ax + Ay + Az, where Ax, Ay, and Az are the components of the vector in the x, y, and z directions.
Unit Vectors
- A unit vector or normalized vector of a vector is defined as: a@ ≡ A / |A|, where A is the magnitude of the vector.
- The unit vector has a magnitude of 1 and indicates only the direction of a vector.
Components of a Vector
- The x-component of a vector A is given by: Ax = Acos θ (with respect to the x-axis).
- The y-component of a vector A is given by: Ay = Asin θ (with respect to the y-axis).
Decomposition of a 2D Vector
- When decomposing a vector, unit vectors provide a useful way to write component vectors.
- The decomposition of a 2D vector A can be written as: A = Ax ı̂ + Ay ȷ̂, where Ax and Ay are the x and y components of the vector, and ı̂ and ȷ̂ are the unit vectors in the x and y directions.
Decomposition of a 3D Vector
- In a 3D rectangular coordinate system, a vector A can be decomposed into three components: Ax, Ay, and Az, along the x, y, and z axes.
- The full decomposition of a 3D vector A can be written as: A = Ax ı̂ + Ay ȷ̂ + Az ̂, where Ax, Ay, and Az are the x, y, and z components of the vector, and ı̂, ȷ̂, and ̂ are the unit vectors in the x, y, and z directions.
This quiz covers the basics of scalars and vectors, including their representations, unit vectors, and vector addition. It's a must-take for physics and math students!
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