Rectangular Vector Form Calculator

Enter Vector Values

The page stays single-column, while the calculator fields shift to three columns on large screens, two on medium screens, and one on mobile.

Vector A: x-component

Vector A: y-component

Vector A: z-component

Vector B: x-component

Vector B: y-component

Vector B: z-component

Scalar multiplier for A

Decimal places (0 to 8)

Example Data Table

Vector A	Vector B	\|A\|	\|B\|	A·B	A×B	Angle	Projection of A on B	A + B
3i - 4j + 12k	-2i + 5j + 1k	13.00	5.48	-14.00	-64i - 27j + 7k	101.34°	0.9333i - 2.3333j - 0.4667k	1i + 1j + 13k

Formula Used

Rectangular vector form: v = xi + yj + zk
Magnitude: |v| = √(x² + y² + z²)
Unit vector: v̂ = v / |v|
Dot product: A·B = x₁x₂ + y₁y₂ + z₁z₂
Cross product: A×B = (y₁z₂ − z₁y₂)i + (z₁x₂ − x₁z₂)j + (x₁y₂ − y₁x₂)k
Angle between vectors: cos θ = (A·B) / (|A||B|)
Scalar projection of A on B: comp_B(A) = (A·B)/|B|
Vector projection of A on B: proj_B(A) = [(A·B)/|B|²]B
Direction cosines: l = x/|v|, m = y/|v|, n = z/|v|
Direction angles: α = cos⁻¹(x/|v|), β = cos⁻¹(y/|v|), γ = cos⁻¹(z/|v|)
Azimuth angle: tan⁻¹(y/x) measured in the xy-plane
Elevation angle: tan⁻¹(z / √(x² + y²))

How to Use This Calculator

Enter the x, y, and z components for Vector A.
Enter Vector B if you want comparisons, projections, dot product, or cross product.
Choose a scalar value when you want the calculator to show a scaled version of Vector A.
Set the number of decimal places for neat output formatting.
Press the calculate button to show the result block below the header and above the form.
Review rectangular form, coordinate form, magnitude, unit vectors, direction cosines, and angles.
Use the CSV button for spreadsheet-ready output.
Use the PDF button for a quick printable summary.

FAQs

1) What is rectangular vector form?

Rectangular vector form writes a vector using perpendicular component directions. In three dimensions, it is usually written as xi + yj + zk, where x, y, and z are the signed component values.

2) Why is magnitude important?

Magnitude gives the vector’s size or length. It is useful for comparing vectors, finding unit vectors, computing projections, and determining the angle between two vectors.

3) What happens if one vector is zero?

A zero vector has no direction, so unit vector, direction cosines, direction angles, and angle-between-vector calculations become undefined. Addition and subtraction still work normally.

4) What is the difference between dot and cross product?

The dot product returns a scalar and measures directional similarity. The cross product returns a vector perpendicular to both input vectors and is useful for area and orientation problems.

5) When should I use projection values?

Use projections when you need the part of one vector acting along another direction. This helps in force resolution, component analysis, navigation, and many geometry problems.

6) What are direction cosines?

Direction cosines are the normalized component ratios of a vector. They show how strongly the vector points along the x, y, and z axes.

7) Can this calculator handle two-dimensional vectors?

Yes. Enter zero for the z-components. The calculator will still compute rectangular form, magnitude, unit vectors, dot product, projections, and the planar direction angle information.

8) What is azimuth versus elevation?

Azimuth is the direction in the xy-plane measured from the positive x-axis. Elevation shows how far the vector tilts upward or downward from that plane.