8. Vector algebra (page 2)

8. Vector algebra


Vector algebra in Cartesian form	Page 2 of 2

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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