Scalars and Vectors

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Questions and Answers

What is the direction angle of a vector 𝐴 = 10.0 N, 30.0° N of E?

60.0° E of N (correct)

30.0° W of N

60.0° W of N

30.0° E of N

When determining the correct direction angle, what should you start with?

The components of the vector

Trigonometric relations

Calculations

A vector diagram (correct)

What is the direction of the vector 𝐴 = 10.0 N, 60.0° E of N?

North of East (correct)

South of East

West of North

East of North

How do you measure the direction angle in the first quadrant?

Counterclockwise from the +x-axis Signup and view all the answers

What is the direction of the vector 𝐴 = 10.0 N, 30.0° W of N?

West of North Signup and view all the answers

How do you measure the direction angle in the second quadrant?

Clockwise from the +x-axis Signup and view all the answers

What is the direction of the vector 𝐴 = 10.0 N, 30.0° E of S?

East of South Signup and view all the answers

What is the x-component of vector A in terms of its magnitude and angle θ?

Acos θ Signup and view all the answers

What is the significance of examining the signs of the components of the vector?

To determine the quadrant of the vector Signup and view all the answers

In which quadrant would a vector have a positive x-component and a negative y-component?

IV Signup and view all the answers

What is the y-component of vector A in terms of its magnitude and angle ϕ?

Asin ϕ Signup and view all the answers

If the x-component of a vector is negative, in which quadrant would it lie?

II or III Signup and view all the answers

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 Signup and view all the answers

What is the direction of the angle measured from the +x-axis?

Clockwise Signup and view all the answers

What is the y-component of vector A in terms of its magnitude and angle θ?

Asin θ Signup and view all the answers

What is the value of the x-component of vector A?

3.75m Signup and view all the answers

What is the magnitude of vector A?

7.1m Signup and view all the answers

What is the purpose of the unit vector of a vector?

To indicate the direction of the vector Signup and view all the answers

What is the magnitude of the unit vector?

1 Signup and view all the answers

What is the result of dividing the perpendicular component of a vector by its magnitude?

+1 or -1 Signup and view all the answers

What is the formula for the unit vector?

a@ = A/A Signup and view all the answers

What is the unit vector for the x-coordinate?

ı̂ Signup and view all the answers

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 Signup and view all the answers

Which of the following is NOT a correct way to write the component vector of A in the x-direction?

A $ ȷ̂ Signup and view all the answers

What is the full decomposition of the vector A?

A = A $ ı̂ + A % ȷ̂ Signup and view all the answers

In a 3D rectangular coordinate system, which unit vector is along the x-axis?

ı̂ Signup and view all the answers

What is the purpose of unit vectors in decomposing a vector?

To provide a useful way to write component vectors Signup and view all the answers

Which of the following is a correct way to write the y-component of a vector?

A % ȷ̂ Signup and view all the answers

What is a characteristic of a scalar quantity?

It can be completely described by a magnitude alone. Signup and view all the answers

What is an example of a vector quantity?

Displacement, 20 m from the origin Signup and view all the answers

What is the equation for the magnitude of a vector 𝐴⃗?

𝐴 = |𝐴⃗| Signup and view all the answers

How is a vector quantity represented in analytic form?

𝐴⃗ = 𝐴𝑎 Signup and view all the answers

What determines the component form of a vector representation?

The coordinate system used. Signup and view all the answers

What is the symbol used to represent a vector quantity?

𝐴⃗ Signup and view all the answers

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.

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Description

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!