vector algebra
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Examples forVectors

Vectors

Compute properties of a vector:

vector {2, -5, 4}

Specify a vector as a linear combination of unit vectors:

vector 3i + 5j
vector 2i - 4j + 3k

Compute the norm of a vector:

norm {12, -5}

Vector Algebra

Do vector computations:

vector (1,3,-1) + (-2,1,6)
7 {1, 0, -2, 1} - 4 {2, -1, 1, -1}
(i + j + k) + (2i - 3j + 8k)

Compute a dot product:

{12, 20} . {16, -5}
(7i-j+3k).(4i-2k)

Compute a cross product:

{1/4, -1/2, 1} cross {1/3, 1, -2/3}
(8i + 3j - k) x (i - j + 2k)

Compute a (scalar) cross product in two dimensions:

(4,1) x (-5,6)

Normalize a vector:

normalize the vector (3, 10)
normalize vector

Convert to another coordinate system:

(1,1,-3) in spherical coordinates

Orthogonality

Check if the set of vectors is orthogonal:

Are (4, -1, -2), (2, -2, 5) and (-9, -24, -6) orthogonal?
Orthogonality (0, 1) and (1, 0)
orthogonality (5, 8, 13, 21) (-12, 5, -6, 1.41) (1.618, 0, 5, 7) (2.72, 7, -9, 3.14)

Find conditions for orthogonality between vectors with symbolic components:

(-a+b, 7) (2, a) orthogonality
Orthogonality (1, 2, 3) (a, b, c) (b, g, f)

Vector Projections

Compute the projection of one vector onto another:

projection vector (2i + 6j) onto (6i + 2j)
projection of vector (-1, 1) onto vector (1, 1)
projection of 2i - 3j + 5k onto 2i - 3j - 5k
projection (3, 0, 4), (0, 5, 12)

Project a vector onto an axis:

project (2, 5) onto the x-axis

Project a vector onto a plane:

What is the projection of the point (3, 4, 5) on the xy-plane?
project vector (6, 5, 4) onto yz-plane

Explore vector projections in higher dimensions:

projection of (2, 4, 10, 5) onto span of {(1, 2, -6, -1), (-2, 7, 4, -12)}

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