Vectors Calculator

Vectors calculator

Vector A = i + j + k

|A|

Unit Vector:

Angle to axis:

x =

y =

z =

Spherical Coordinate:

r =

θ =

Φ =

Vector B = i + j + k

|B|

Unit Vector:

Angle to axis:

x =

y =

z =

Spherical Coordinate:

r =

θ =

Φ =

A + B
A − B
A ⋅ B
C = A ⨉ B
\|C\|
Angle Between Vectors:


Vectors Definition	Vectors Cross Product	Vectors functions and derivation
Vectors Addition	Integration of Vectors	Vector spherical cylindrical coordinates
Vectors Dot Product

Vector definition

Vectors definition

A vector V is represented in three-dimensional space in terms of the sum of its three mutually perpendicular components.

Where i, j and k are the unit vector in the x, y and z directions respectively and has magnitude of one unit.

The scalar magnitude of V is:

Let V be any vector except the 0 vector, the unit vector q in the direction of V is defined by:

A set of vectors for example {u, v, w} is linearly independent if and only if the determinant D of the vectors is not 0.

Two vectors V and Q are said to be parallel or proportional, when each vector is a scalar multiple of the other and neither is zero.

Vectors addition (A ± B)

Two vectors A and B may be added to obtain their resultant or sum A + B, where the two vectors are the two legs of the parallelogram.

Vectors addition obey the following laws:

Commutative law:	A + B = B + A
Associative law:	A + (B + C) = (A + B) + C
(c and d are any number)	c(dA) = (cd)A
Distributive law:	(c + d)A = cA + dA c(A + B) = cA + cB

The subtraction of a vector is the same as the addition of a negative vector

A − B = A + (−B)

1 · A = A

0 · A = 0

(− 1)A = −A

If A and B are two vectors then the following relations are true:

Example: find the diagonal length and the unit vector of a rectangle defined by the vectors A = 4i and B = 3j in the R direction, also find angle θ by the dot product.

Solution:	Diagonal vector:	R = A + B = 4i + 3j

	Diagonal length:
	The unit vector in the R direction is:
	dot product A · R in order to find cosθ:

Dot or scalar product (A · B)

Dot or scalar product (A · B)

The dot or scalar product of two vectors A and B is defined as:

(The result is a scalar value)

θ is the angle between the two vectors

From the definition of the dot product, it follows that:

Thus:

And

Two vectors are perpendicular when their dot product is:

A · B = 0

The angle θ between two vectors A and B is:

Where l, m and n stands for the respective direction cosines of the vectors. It is also observed that two vectors are perpendicular when their direction cosines obey the relation:

Where:

The dot product satisfies the following laws:

Commutative law:	A · B = B · A
Distributive law:	A · (B + C) = A · B + A · C
If m is a scalar:	m(A · B) = (mA) · B = A · (mB) = (A · B)m
Other relations:	(A · B)C ≠ A(B · C)
	A · A = \|A\|²
Parallel vectors when:	A · B = ±\|A\|\|B\|

Example: given two vectors A = 2i + 2j − k and B = 6i − 3j + 2k. Find the vectors dot product and the angle between the vectors.

Solution The dot product is:	A · B = 2 · 6 − 2 · 3 − 1 · 2 = 4
Magnitude of A and B are:

The angle between the vectors is:

Cross or vector product (A✕ B)

The cross or vector product of two vectors A and B is defined as:

(The result is a vector)

n - unit vector whose direction is perpendicular to vectors A and B.

Note: the direction of A ✕ B is normal to the plane defined by A and B and is pointing according to the right-hand screw rule.

From the definition of the cross product the following relations between the vectors are apparent:

The vector product is written as:

This expression may be written as a determinant:

The cross product obeys the following laws:

The commutative law does not hold for cross product because:

A ✕ B = − (B ✕ A) and A ✕ (B ✕ C) ≠ (A ✕ B) ✕ C

Distributive law:	A ✕ (B + C) = A ✕ B + A ✕ C
	(A + B) ✕ C = (A ✕ C) + (B ✕ C)
If m is a scalar, then:	m(A ✕ B) = (mA) ✕ B = A ✕ (mB) = (A ✕ B)m

Triple scalar product is defined as the determinant:

(A ✕ B) ✕ C = -C ✕ (A ✕ B) = C ✕ (B ✕ A)

Other relations:	A · (A ✕ C) = 0
	A ✕ (B ✕ C) + B ✕ (C ✕ A) + C ✕ (A ✕ B) = 0
	A ✕ (B ✕ C) =(A · C)B − (A · B)C
	(A ✕ B) ✕ C = (A · C)B − (B · C)A
	(A ✕ B) · (C ✕ D) = (A · C)(B · D) − (A · D)(B · C)

Example: find the area of the triangle ABC and the equation of the plane passing through points A, B and C if the points coordinates are: A(1, -2, 3), B(3, 1, 2) and C(2, 3, -1).

Solution: the vectors presentations are:

Vector
Vector

The area of the parallelogram is:
The area of triangle ABC is:

The normal to the plane vector is:
The plane equation will be accordingly:

Vectors functions and derivation

The derivative of a vector P according to a scalar variable t is:

The derivative of the sum of two vectors is:

The derivative of the product of a vector P and a scalar u(t)according to t is:

The derivative of two vectors dot product:

For the cross product the derivative is:

Gradient, If ϕ is a scalar function defined by ϕ=f(x,y,z), we define the gradient of ϕ, that is a vector in the n-direction and represents the maximum space rate of change of ϕ.

The del operator:
Gradient operator:

divergence, when the vector operator ᐁ is dotted into a vector V, the result is the divergence of V.

Note that:

Curl, When the vector operator ᐁ is crossed into a vector V, the result is the curl of V.

Laplacian operator is the dot product of the vector ᐁ into itself gives the scalar operator known as Laplacian operator.

Laplacian operator

Laplace's equation

Other relations of the ᐁ operator.

ᐁ ✕ (ᐁΦ ) = 0

ᐁ· (ᐁ✕A ) = 0

ᐁ✕(ᐁ✕A ) = ᐁ(ᐁ·A)−ᐁ²A

ᐁ(Φ + ψ ) = ᐁΦ + ᐁψ

ᐁ·(A + B ) = ᐁ·A + ᐁ·B

ᐁ✕(A + B ) = ᐁ✕A + ᐁ✕B

ᐁ·(ΦA ) = (ᐁΦ)·A + Φ(ᐁ·A )

ᐁ✕(ΦA) = (ᐁΦ)✕A + Φ(ᐁ✕A )

ᐁ· (A✕B ) = B · (ᐁ✕A ) − A·(ᐁ✕B )

ᐁ✕(A✕B ) = (B ·ᐁ)A − B(ᐁ·A ) − (A·ᐁ)B + A(ᐁ·B )

ᐁ(A·B ) = (B·ᐁ)A + (A·ᐁ)B + B✕(ᐁ✕A ) A✕(ᐁ✕B )

Vectors integration

If V is a function of x, y, and z and an element of volume is dv = dx dy dz, the integral of V over the volume may be written as the vector sum of the three integrals of its components: