Precalc Calculator

Vector Operations Calculator

This calculator helps you perform common operations on 3D vectors, including addition, subtraction, scalar multiplication, dot product, and magnitude calculation. Vectors are fundamental in physics, engineering, and computer graphics, representing quantities with both magnitude and direction.

Understanding Vector Operations

In mathematics and physics, a vector is a quantity that has both magnitude and direction. It is often represented as an arrow, where the length of the arrow is the magnitude and the way the arrow points is the direction. In a 3D Cartesian coordinate system, a vector can be described by its components along the x, y, and z axes, typically written as <x, y, z>.

Vector Addition

Vector addition combines two vectors to produce a resultant vector. Geometrically, if you place the tail of the second vector at the head of the first, the resultant vector goes from the tail of the first to the head of the second. Algebraically, you add the corresponding components of the vectors:

If A = <A_x, A_y, A_z> and B = <B_x, B_y, B_z>, then A + B = <A_x + B_x, A_y + B_y, A_z + B_z>.

Example: If A = <3, 4, 5> and B = <1, 2, 3>, then A + B = <3+1, 4+2, 5+3> = <4, 6, 8>.

Vector Subtraction

Vector subtraction is similar to addition, but you subtract the corresponding components. Subtracting vector B from vector A is equivalent to adding A to the negative of B (a vector with the same magnitude but opposite direction).

A – B = <A_x – B_x, A_y – B_y, A_z – B_z>.

Example: If A = <3, 4, 5> and B = <1, 2, 3>, then A – B = <3-1, 4-2, 5-3> = <2, 2, 2>.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single number). This operation scales the magnitude of the vector by the scalar value. If the scalar is positive, the direction remains the same; if negative, the direction reverses.

If A = <A_x, A_y, A_z> and s is a scalar, then sA = <sA_x, sA_y, sA_z>.

Example: If A = <3, 4, 5> and s = 2, then 2A = <2*3, 2*4, 2*5> = <6, 8, 10>.

Dot Product (Scalar Product)

The dot product of two vectors results in a scalar value. It is a measure of how much two vectors point in the same direction. If the dot product is positive, the vectors generally point in the same direction; if negative, they generally point in opposite directions; if zero, they are orthogonal (perpendicular).

A · B = A_xB_x + A_yB_y + A_zB_z.

Example: If A = <3, 4, 5> and B = <1, 2, 3>, then A · B = (3*1) + (4*2) + (5*3) = 3 + 8 + 15 = 26.

Magnitude of a Vector

The magnitude (or length) of a vector is a scalar value representing its size. It is calculated using the Pythagorean theorem in 3D space.

|A| = √(A_x² + A_y² + A_z²)

Example: If A = <3, 4, 5>, then |A| = √(3² + 4² + 5²) = √(9 + 16 + 25) = √50 &approx; 7.071.

Vector A + B:

Vector A – B:

Scalar (' + scalarValue.toFixed(3) + ') * Vector A:

Dot Product (A · B):

Magnitude of Vector A (|A|):

Magnitude of Vector B (|B|):