Representing Vectors in Physics

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What is the difference between scalar and vector quantities?

Scalar quantities have only magnitude, while vector quantities have both magnitude and direction.

How can a vector be represented?

A vector can be represented by an arrow over the variable (e.g. ā) or by a bold face (e.g. a).

What are the three coordinate systems used to specify the magnitude and direction of a vector?

The three coordinate systems are Cartesian (rectangular), cylindrical, and spherical.

What is the magnitude of a vector, and how is it represented?

The magnitude of a vector is its length, represented by the symbol r, and is often calculated using the Pythagorean theorem. Signup and view all the answers

What is the difference between a real number and a complex number in the context of vectors?

A real number represents a single value (e.g. 10s), while a complex number represents a composite of two values (e.g. 15+3i A). Signup and view all the answers

What is the significance of unit vectors in vector algebra?

Unit vectors are used to specify the direction of a vector and have a magnitude of one. Signup and view all the answers

How do you determine the direction of a vector in 2D or 3D space?

The direction of a vector is given by the angle it makes with the positive x-axis. Signup and view all the answers

What are the three components of a point in 3D space?

(x, y, z) Signup and view all the answers

How can a vector in 2D (or 3D) space be decomposed?

A vector can be decomposed into its component vectors, which lie along the coordinate axes. Signup and view all the answers

How are the component vectors of a vector related to its magnitude and angle with the positive x-axis?

The component vectors are given by A cos(θ) and A sin(θ), where A is the magnitude and θ is the angle. Signup and view all the answers

What is the relationship between the components of a vector and its magnitude?

The magnitude of a vector is related to its components through the Pythagorean theorem. Signup and view all the answers

How can the magnitude of an n-dimensional vector be evaluated?

The magnitude of an n-dimensional vector can be evaluated using the Euclidean norm. Signup and view all the answers

What is the purpose of unit vectors in a coordinate system?

To specify a given direction in space. Signup and view all the answers

How can a vector be represented using unit vectors?

As a sum of its components with the help of unit vectors. Signup and view all the answers

What is the notation for representing scalar components of a vector?

A_x, A_y, A_z Signup and view all the answers

How can an arbitrary unit vector be obtained for a given vector?

By dividing the given vector by its magnitude. Signup and view all the answers

How do you find the magnitude of a displacement vector if you know the initial and final positions of an object?

By using the distance formula, √((x2 - x1)^2 + (y2 - y1)^2). Signup and view all the answers

What is the difference between the magnitude of a displacement vector and the distance traveled by an object?

The magnitude of a displacement vector is the shortest distance between the initial and final positions, while the distance traveled is the total path length. Signup and view all the answers