Glossary
2D Vector Components
The horizontal (x) and vertical (y) parts that define a vector's movement in a 2D plane, typically written as <a, b>.
Example:
A vector representing a movement 3 units right and 4 units up has 2D vector components of <3, 4>.
Direction
The orientation of a vector in space, indicating where it points.
Example:
If a force pushes an object upwards, that's the direction of the force vector.
Dot Product
An operation that takes two vectors and returns a scalar, indicating the extent to which the vectors point in the same direction.
Example:
Calculating the work done by a force over a displacement involves the dot product of the force and displacement vectors.
Law of Cosines
A formula relating the sides of a triangle to the cosine of one of its angles, useful for solving non-right triangles when given three sides or two sides and the included angle.
Example:
To find the length of the third side of a triangle when you know two sides and the angle between them, apply the Law of Cosines.
Law of Sines
A formula relating the sides of a triangle to the sines of their opposite angles, useful for solving non-right triangles when given certain angle-side combinations.
Example:
If you know two angles and one side of a triangle, you can use the Law of Sines to find the other sides.
Linear Combination
The expression of a vector as the sum of scalar multiples of other vectors, typically using standard unit vectors.
Example:
Any 2D vector <a, b> can be expressed as ai + bj, which is a linear combination of the standard unit vectors.
Magnitude
The length or size of a vector, representing its quantity or intensity.
Example:
The speed of a car, like 60 mph, is the magnitude of its velocity vector.
Orthogonal
A term describing two vectors that are perpendicular to each other, meaning their dot product is zero.
Example:
The x-axis and y-axis are orthogonal because they meet at a 90-degree angle.
Projections
The components of a vector onto the x and y axes, representing how much of the vector lies along each axis.
Example:
The shadow a vector casts on the x-axis is its x-axis projection.
Scalar
A quantity that only has magnitude, without direction.
Example:
Temperature (e.g., 25°C) is a scalar quantity because it doesn't have a direction.
Scalar Multiplication
The operation of multiplying a vector by a constant (scalar), which scales its magnitude and can reverse its direction if the scalar is negative.
Example:
If vector v represents a 5 mph walk north, then 2v represents a 10 mph walk north, illustrating scalar multiplication.
Standard Unit Vectors (i, j)
Specific unit vectors along the positive x-axis (i = <1, 0>) and positive y-axis (j = <0, 1>) in a 2D coordinate system.
Example:
The vector <3, 5> can be written as 3i + 5j using standard unit vectors.
Triangle Inequality Theorem
States that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Example:
You cannot form a triangle with side lengths 2, 3, and 6 because 2 + 3 is not greater than 6, violating the Triangle Inequality Theorem.
Trigonometric Components
The x and y components of a vector found using its magnitude and the angle it makes with the x-axis, typically using cosine for x and sine for y.
Example:
A vector with magnitude 10 at a 30-degree angle has x-component 10cos(30) and y-component 10sin(30), which are its trigonometric components.
Unit Vector
A vector that has a magnitude of 1, used to indicate direction without conveying magnitude.
Example:
To find the direction of a force without considering its strength, you'd use its unit vector.
Vector
An arrow-like quantity possessing both magnitude (length) and direction, used to represent movement, force, or other directional quantities.
Example:
A plane's velocity of 500 mph east is a vector.
Vector Addition
The process of combining two or more vectors to find a single resultant vector, often done component-wise or using the head-to-tail rule.
Example:
Combining a vector representing walking 3 miles east with a vector representing walking 4 miles north results in a new displacement vector through vector addition.
Zero Vector
A vector with zero magnitude, where its tail and head are at the same point, represented as <0, 0>.
Example:
If you start and end at the exact same spot, your displacement is the zero vector.