Introduction to Vectors

<p>Three-dimensional space (B)</p> Signup and view all the answers

<p>All possible vectors with defined vector addition and scalar multiplication (C)</p> Signup and view all the answers

<p>Vectors have both magnitude and direction. (C)</p> Signup and view all the answers

<p>$ ext{A} = A_x extbf{i} + A_y extbf{j}$ (C)</p> Signup and view all the answers

<p>By placing the tail of one vector at the head of the other. (D)</p> Signup and view all the answers

<p>The vector's direction changes by 180 degrees. (D)</p> Signup and view all the answers

<p>Using the Pythagorean theorem: $A = ext{sqrt}(A_x^2 + A_y^2)$ (A)</p> Signup and view all the answers

<p>A vector with a magnitude of 1. (C)</p> Signup and view all the answers

<p>Physics, engineering, and computer graphics. (D)</p> Signup and view all the answers

<p>$ ext{A}_x$ and $ ext{A}_y$ (D)</p> Signup and view all the answers

The product of two vectors that results in a scalar value. Geometrically, it's the product of the magnitudes of the vectors multiplied by the cosine of the angle between them.

A mathematical operation on two vectors in three-dimensional space that produces a new vector perpendicular to both input vectors. Its magnitude is the product of the magnitudes of the vectors multiplied by the sine of the angle between them.

A set of all possible vectors with specific rules for vector addition and scalar multiplication. These spaces allow us to study the properties and behaviors of vectors in more abstract ways.

In the dot product of two vectors, it's the product of the magnitudes of the two vectors multiplied by the cosine of the angle between them. It provides a measure of how much the vectors align.

The cross product of two vectors is the product of their magnitudes multiplied by the sine of the angle between them. It represents the area of the parallelogram defined by the two vectors.

A mathematical object possessing both magnitude (size) and direction.

Representing a vector as an arrow, where the length of the arrow signifies magnitude and the direction of the arrow denotes the vector's direction.

Breaking a vector down into its components along different axes. For example, a vector in a 2D plane can be decomposed into horizontal (x) and vertical (y) components.

Vectors can be added or subtracted using the parallelogram law or by adding or subtracting their components.

Multiplying a vector by a scalar (a number) scales the vector's magnitude. The direction remains the same if the scalar is positive, and it changes by 180 degrees if the scalar is negative.

The magnitude (or length) of a vector $ \vec{A}$ is denoted by $|| \vec{A}||$ or A. For a 2D vector $ \vec{A} = A_x \hat{i} + A_y \hat{j}$, the magnitude is calculated using the Pythagorean theorem: $A = \sqrt{A_x^2 + A_y^2}$.

A unit vector has a magnitude of 1. Unit vectors are used to represent directions without specifying magnitude. They are often denoted with a 'hat' symbol, like $ \hat{i}$, $ \hat{j}$, and $ \hat{k}$ for the standard coordinate axes in 2D and 3D.

Vectors are fundamental in various fields, including physics (displacement, velocity, acceleration, force), engineering (forces, moments), and computer graphics (movement, transformations). They allow a concise way to represent and manipulate quantities with both magnitude and direction.