School of Mathematics and Statistics, The University of Sydney
 8. Vector algebra
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Glossary
Examples

Vector algebra in Cartesian form

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We discuss properties of the two operations, addition of vectors and multiplication of a vector by a scalar, but now for vectors in Cartesian form.

Suppose vectors v1 and v2 are written in Cartesian form,

v  =  x i + x j + x k,  and  v  =  y i + y j + y k.   1    1     2    3            2    1    2     3

Equality of Vectors
Vectors v1 and v2 are equal if and only if their components are equal, that is, if and only if x1 = y1, x2 = y2 and x3 = y3.
The Negative of a Vector
The negative of the vector v1 = x1i + x2j + x3k is the vector
- v1 = - x1i- x2j - x3k.

Multiplication by a Scalar
If s is any scalar, then
sv1 = s(x1i + x2j + x3k) = sx1i + sx2j + sx3k.

Addition of Vectors
v1+v2  = (x1i+x2j+x3k)+(y1i+y2j+y3k)     = (x1+y1)i+(x2+y2)j+(x3+y3)k.

The Zero Vector
The zero vector is the vector 0 = 0i + 0j + 0k.
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