Question 1

What is vector addition in mathematics?

Accepted Answer

Vector addition is the process of combining two or more vectors to produce a resultant vector. It involves adding corresponding components of the vectors. For vectors A = (ax, ay, az) and B = (bx, by, bz), the sum R = A + B = (ax + bx, ay + by, az + bz). Vector addition follows the commutative and associative properties.

Question 2

How do you calculate vector magnitude?

Accepted Answer

Vector magnitude is calculated using the Pythagorean theorem extended to multiple dimensions. For a vector V = (x, y, z), the magnitude |V| = √(x² + y² + z²). For 2D vectors, it's |V| = √(x² + y²). The magnitude represents the length or size of the vector from its tail to its head.

Question 3

How do you find vector direction?

Accepted Answer

Vector direction is typically expressed as an angle from a reference axis (usually the positive x-axis). For a 2D vector V = (x, y), the direction angle θ = tan⁻¹(y/x). The angle is measured in degrees or radians. For 3D vectors, direction is often expressed using spherical coordinates or direction cosines.

Question 4

What is the dot product of vectors?

Accepted Answer

The dot product (scalar product) of two vectors A and B is a scalar quantity calculated as A · B = ax·bx + ay·by + az·bz. The dot product measures the projection of one vector onto another and is related to the angle between vectors: A · B = |A||B|cos(θ), where θ is the angle between the vectors.

Question 5

What is the cross product of vectors?

Accepted Answer

The cross product (vector product) of two vectors A and B is a vector perpendicular to both A and B. For 3D vectors, A × B = (ay·bz - az·by, az·bx - ax·bz, ax·by - ay·bx). The magnitude of the cross product is |A × B| = |A||B|sin(θ), where θ is the angle between the vectors.

Question 6

How are vectors used in physics?

Accepted Answer

Vectors are fundamental in physics for representing quantities with both magnitude and direction. Common applications include: force vectors in mechanics, velocity and acceleration vectors in kinematics, electric and magnetic field vectors in electromagnetism, and displacement vectors in navigation and mapping.

Question 7

What is the difference between 2D and 3D vectors?

Accepted Answer

2D vectors have two components (x, y) and exist in a plane, while 3D vectors have three components (x, y, z) and exist in three-dimensional space. 3D vectors can undergo cross product operations, while 2D vectors cannot. The mathematical operations are similar, but 3D vectors provide more spatial information and complexity.

Question 8

How do you add vectors graphically?

Accepted Answer

To add vectors graphically using the head-to-tail method: 1) Draw the first vector from the origin, 2) Draw the second vector starting from the head of the first vector, 3) The resultant vector is drawn from the tail of the first vector to the head of the second vector. This method works for any number of vectors.

Question 9

What are unit vectors?

Accepted Answer

Unit vectors are vectors with magnitude equal to 1. They are used to indicate direction only. Common unit vectors are î (x-direction), ĵ (y-direction), and k̂ (z-direction) in 3D space. Any vector can be expressed as a scalar multiple of unit vectors: V = vxî + vyĵ + vzk̂.

Question 10

How do you calculate the angle between two vectors?

Accepted Answer

The angle θ between two vectors A and B can be calculated using the dot product formula: cos(θ) = (A · B)/(|A||B|). Therefore, θ = cos⁻¹((A · B)/(|A||B|)). This formula works for both 2D and 3D vectors and gives the angle in radians or degrees depending on your calculator settings.

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