A vector in the plane is a mathematical object with magnitude and direction, represented by an arrow from one point to another in the two-dimensional plane.
How is a vector in the plane typically represented?
A vector in a plane is typically represented by coordinates (x, y) which describe its position relative to the origin of a Cartesian coordinate system.
What is the standard form of a vector in 2-dimensional space?
The standard form of a vector in 2-dimensional space is expressed as v = ai + bj, where 'a' and 'b' are components of the vector along the x and y axes, respectively.
How are vectors added together?
Vectors are added together by adding their respective components: if v = (a, b) and u = (c, d), then v + u = (a+c, b+d).
What is the difference between vectors and scalars?
Vectors have both magnitude and direction, while scalars have only magnitude.
What is the magnitude of a vector?
The magnitude of a vector (a, b) is calculated as the square root of the sum of the squares of its components: √(a² + b²).
How can you find the direction of a vector?
The direction of a vector is often given by the angle it makes with the positive x-axis, calculated using the arctangent function: angle = atan2(b, a).
What is a unit vector?
A unit vector is a vector with a magnitude of 1, often used to indicate direction.
How is a vector multiplied by a scalar?
To multiply a vector by a scalar, multiply each component of the vector by the scalar: if v = (a, b) and k is a scalar, then kv = (ka, kb).
What does it mean for two vectors to be equal?
Two vectors are equal if they have the same magnitude and direction, meaning their corresponding components are equal.
How can you determine if two vectors are parallel?
Two vectors are parallel if they are scalar multiples of each other, meaning one vector can be expressed as a constant times the other.
What is the dot product of two vectors and how is it calculated?
The dot product of two vectors v = (a, b) and u = (c, d) is a scalar calculated as (a * c) + (b * d).
What geometric information does the dot product provide?
The dot product provides information about the angle between two vectors. A dot product of zero indicates that the vectors are perpendicular.
What is the cross product of two vectors in the plane?
In two-dimensional space, the cross product is not directly defined like in three dimensions, but the concept can be related by the magnitude of the determinant of a 2x2 matrix formed by their components.
How is vector subtraction performed?
Vector subtraction is performed by subtracting the corresponding components of the vectors: if v = (a, b) and u = (c, d), then v - u = (a-c, b-d).