Vector calculator, dot product, cross product and angle
Norms, sum, dot product, cross product, angle, projection and a collinearity test, in the plane as in space.
Result
7
- Calculation
- u · v = uₓvₓ + uyvy + uzvz = 7
- Angle between u and v
- θ = arccos(u · v / (‖u‖‖v‖)) = arccos(7 / (4.12311 × 3.16228)) = 57.5288°
- Calculation
- u × v = (u_yv_z − u_zv_y, u_zvₓ − uₓv_z, uₓv_y − u_yvₓ) = (0, 0, 11)
- Parallelogram area
- 11
Each vector
- Norm of u
- 4.12311
- Norm of v
- 3.16228
- Unit vector of u
- (0.970 143, 0.242 536)
- Unit vector of v
- (0.316 228, 0.948 683)
What links u and v
- Angle between u and v
- 57.5288°
- Distance between the endpoints
- 3.60555
Combinations
- Sum u + v
- (5, 4)
- Difference u − v
- (3, −2)
- Product 2 · u
- (8, 2)
Decomposition of u onto v
- Scalar projection of u onto v
- 2.21359
- Projection of u onto v
- (0.7, 2.1)
- Component of u orthogonal to v
- (3.3, −1.1)
Notes (2)
The two vectors make an angle of 57.5288°, neither zero, nor right, nor straight: they are neither collinear nor orthogonal.
In the plane, u × v lies along z; that component is the determinant, positive counterclockwise.
Calculation, assumptions and sources
OverviewFor two vectors of the plane or of space: norms and unit vectors, sum, difference, product by a scalar, dot product, cross product (determinant in the plane), parallelogram area, angle in degrees and radians, projection of one onto the other (parallel and orthogonal parts), collinearity and orthogonality tests.
For two vectors of the plane or of space: norms and unit vectors, sum, difference, product by a scalar, dot product, cross product (determinant in the plane), parallelogram area, angle in degrees and radians, projection of one onto the other (parallel and orthogonal parts), collinearity and orthogonality tests. For a zero vector, direction and angle are marked undefined.
Method & formulas‖u‖ = √(uₓ² + u_y² + u_z²)
The dot product carries the cosine of the angle, the cross product its sine. Lagrange’s identity ties them together: ‖u × v‖² + (u · v)² = ‖u‖²‖v‖².
- ‖u‖
- norm of the vector, its length.
- Unit vector
- u divided by its norm: same direction, length 1. Undefined for the zero vector.
- u · v
- dot product, a number. Positive, both vectors point the same way; negative, opposite ways; exactly zero when they are perpendicular.
- u × v
- cross product, a vector perpendicular to both, exactly zero when they are collinear. Its norm is the area of the parallelogram built on u and v, twice that of the triangle.
- Determinant
- in the plane, the z component of the cross product: uₓv_y − u_yvₓ. Its sign gives the sense of rotation.
- Projection
- the part of u carried by the direction of v. The rest, u minus that part, is orthogonal to v.
Validity domainThe collinearity and orthogonality tests apply a relative tolerance to the cosine and sine of the angle, not to the products: their answer does not depend on the size of the vectors.
The collinearity and orthogonality tests apply a relative tolerance to the cosine and sine of the angle, not to the products: their answer does not depend on the size of the vectors. An angle of a hundredth of a degree is not taken for collinearity. In dimension 2, the z components are 0.