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05. Vector Equations

Dated: 11-03-2025

Column Vector

A matrix1 with only one column is called column vector or simply a vector.2

\[ \vec v = \begin{bmatrix} 2 & 3 & 5 \end{bmatrix}^T = \begin{bmatrix} 2 \\ 3 \\ 5 \end{bmatrix} \]

Vectors in \(\mathbb R^2\)

If \(\mathbb R\) is the set3 of all real numbers then the set3 of all vectors2 with two entries1 is denoted by \(\mathbb R^2 = \mathbb R \times \mathbb R\).

\[ \vec v = \begin{bmatrix} 3 & -1 \end{bmatrix}^T = \begin{bmatrix} 3 \\ -1 \end{bmatrix} \in \mathbb R^2 \]

The entries1 of the vectors2 are assumed to be the elements of a set3 called a field. It is denoted by \(F\).

Algebra of Vectors

Equality of Vectors in \(\mathbb R^2\)

Two vectors2 in \(\mathbb R^2\) are equal if and only if their corresponding entries are equal.

\[ \vec u = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} , \quad \vec v = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \in \mathbb R^2 \implies \vec u = \vec v \iff u_1 = v_1 \land u_2 = v_2 \]

\(\vec v = \begin{bmatrix}x \\ y\end{bmatrix}\) in \(\mathbb R^2\) are just order pairs \((x, y)\) representing points relative to the origin.

Addition of Vectors

Given two vectors2 \(\vec u\) and \(\vec v\) in \(\mathbb R^2\), their sum is the vector2 \(\vec v + \vec u\) obtained by adding corresponding entries of the vectors2 \(\vec u\) and \(\vec v\), which is also in \(\mathbb R^2\).

\[ \vec u = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}, \quad \vec v = \begin{bmatrix} v_1\\ v_2 \end{bmatrix} \in \mathbb R^2 \quad \vec u + \vec v = \begin{bmatrix} u_1\\ u_2 \end{bmatrix} + \begin{bmatrix} v_1\\ v_2 \end{bmatrix} = \begin{bmatrix} u_1 + v_1\\ u_2 + v_2 \end{bmatrix} \in \mathbb R^2 \]

Scalar Multiplication of a Vector

\[ c \times \vec v = c \begin{bmatrix} v_1\\ v_2 \end{bmatrix} = \begin{bmatrix} cv_1\\ cv_2\\ \end{bmatrix} \]

Here \(\vec v\) is a vector2 and \(c\) is a scalar.

Geometric Descriptions of \(\mathbb R^2\)

The \(\mathbb R^2\) can be regarded as a plane4 consisting of ordered pairs \((x, y)\).
These \((x, y)\) represent a point as well as a vector2 \(\begin{bmatrix}x\\ y\end{bmatrix}\).

Geometric Descriptions of \(\mathbb R^3\)

The \(\mathbb R^3\) can be regarded as a three dimensional coordinate space consisting of points \((x, y, z)\) or alternatively, vectors2 \(\begin{bmatrix}x\\ y\\ z\end{bmatrix}\).

Vectors in \(\mathbb R^n\)

If \(n\) is a positive integer, \(\mathbb R^n\) denotes the collection of all ordered pairs \((u_1, u_2, \ldots, u_n)\) or vectors2

\[ \vec u = \begin{bmatrix} u_1 & u_2 & \ldots & u_n \end{bmatrix}^T \]
Null Vector2

The vector2 with all entries1 as \(0\) is called a null vector.2

Algebraic Properties of \(\mathbb R^n\)

For all \(\vec u, \vec v, \vec w\) in \(\mathbb R^n\) and all scalars \(c\) and \(d\).

  • \(\vec u + \vec v = \vec v + \vec u\) (Commutative)
  • \((\vec u + \vec v) + \vec w = \vec u + (\vec v + \vec w)\) (Associative)
  • \(\vec u + \vec 0 = \vec 0 + \vec u\) (Additive Identity)
  • \(\vec u + (-\vec u) = (- \vec u) + \vec u = 0\) (Additive Inverse)
  • \(c(\vec u + \vec v) = (c\vec u + c\vec v)\) (Scalar Distribution over Vector Addition)
  • \(\vec u(c + d) = c\vec u + d\vec u\) (Vector Distribution over Scalar Addition)
  • \(c(d\vec u) = (cd)\vec u\)
  • \(1 \vec u = \vec u\) (Multiplicative Identity)

Linear Combination

Given vectors2 \(\vec v_1, \vec v_2, \ldots, \vec v_n\) in \(\mathbb R^n\) and given the scalars \(c_1, c_2, \ldots, c_n\) the vector2 defined by

\[\vec y = c_1\vec v_1 + c_2 \vec v_2 + \cdots + c_n \vec v_n\]

is called a linear combination of \(\vec v_1, \vec v_2, \ldots, \vec v_n\) with weights \(c_1, c_2, \ldots, c_n\).

Spanning Set

The set3 of all the linear combinations of \(\vec v_1, \vec v_2, \ldots, \vec v_n \in \mathbb R^n\) is the subset3 of \(\mathbb R^n\) spanned (or generated) by those vectors.2

If we want to check if \(\vec b \in S\) (where \(S\) is our span), we can check if either of the following has a solution or not.

  • \(a_1 \vec v_1 + a_2 \vec v_2 + \cdots + a_n \vec v_n = \vec b\)
  • \(\begin{bmatrix}\vec v_1 & \vec v_2 & \cdots & \vec v_n & \vec b\end{bmatrix}\)

A Geometric Description of Span \(\{\vec v\}\)

Let \(\vec v\) be a vector2 in \(\mathbb R^3\) then its span is all linear combinations of \(a\vec v\) where \(a\) is a constant. This results into a line5 passing through \(\vec 0\) and \(\vec v\).

A Geometric Description of Span \(\{\vec U, \vec v\}\)

Let \(\vec v\) and \(\vec u\) be two vectors2 in \(\mathbb R^3\) and \(\vec u\) not be a multiple of \(\vec v\) then their span is all linear combinations consisting of \(\vec v\) and \(\vec u\). This includes the lines5 passing through \(\vec u\) and \(\vec 0\), \(\vec v\) and \(\vec 0\), resulting into a whole plane.4

Linear Combinations in Applications

Example

A manufacturing company manufactures 2 products.
For one dollar's worth of product B, company spends \(\$0.45\) on materials, \(\$0.25\) on labor, \(\$0.15\) on overhead.
For one dollar's worth of product C, company spends \(\$0.40\) on materials, \(\$0.30\) on labor, \(\$0.15\) on overhead.
Let \(\vec b\) and \(\vec c\) represent costs per dollar on the products then

\[ \vec b = \begin{bmatrix} .45 \\ .25 \\ .15 \end{bmatrix} \]
\[ \vec c = \begin{bmatrix} .40 \\ .30 \\ .15 \end{bmatrix} \]
  1. What economic interpretation can be given to the vector2 \(100 \vec b\).
  2. Suppose the company wishes to manufacture \(x_1\) dollars worth of product B and \(x_2\) dollars worth of product C. Give a vector2 that describes the various costs the company will have (for materials, labor and overhead).

Solution

First part
\[ \vec b = \begin{bmatrix} .45 \\ .25 \\ .15 \end{bmatrix} \]
\[ 100 \vec b = 100 \begin{bmatrix} .45 \\ .25 \\ .15 \end{bmatrix} \]
\[ = \begin{bmatrix} 45 \\ 25 \\ 15 \end{bmatrix} \]

The vector2 \(100 \vec b\) represents a list of various costs for producing \(\$100\) worth of product B, namely, \(\$45\) for materials, \(\$25\) for labor, \(\$15\) for overhead.

Second part

The costs of manufacturing \(x_1\) dollars worth of B are given by the vector2 \(x_1 \vec b\) and the costs of manufacturing \(x_2\) dollars worth of C are given by \(x_2 \vec c\). Hence the total costs for both products are given by the vector2 \(x_1 \vec b + x_2 \vec c\).

Vector Equation of a line

mth501_e_5_1.svg

Through vector addition,2 we can define \(\vec v\).

\[\vec {x_0} + \vec v = \vec{x_1}\]
\[\vec v = \vec{x_1} - \vec{x_0}\]

Therefore, the vector2 \(\vec{x_1} - \vec{x_0}\) and the vector2 \(t \vec v\) where \(t\) is a scalar, are parallel vectors.2
This can be written as

\[t \vec v = \vec{x_1} - \vec{x_0} \quad \text{where } (- \infty < t < \infty)\]

where \(t\) is also called the parameter.
This equation is called the vector equation of a line, through \(x_0\) and parallel to \(\vec v\).

Parametric Equation of a line in \(\mathbb R^2\)

Let \(\vec x = (x, y) \in \mathbb R^2\) be a general point of the line5 passing through \(\vec {x_0} = (x_0, y_0) \in \mathbb R^2\) which is parallel to \(\vec v = (a, b) \in \mathbb R^2\).

\[(x, y) - (x_0, y_0) = t(a, b)\]
\[\implies (x - x_0, y - y_0) = (ta, tb)\]
\[\implies x = x_0 + at, \quad y = y_0 + bt \quad (-\infty < t < \infty)\]

These are called parametric equations of a line in \(\mathbb R^2\).

Parametric Equation of a line in \(\mathbb R^2\)

Similarly, the parametric equations of a line in \(\mathbb R^3\) are

\[\implies x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct \quad (-\infty < t < \infty)\]

Vector Equation of a Plane

mth501_e_5_2.svg

\(\vec x - \vec {x_0} = t_1 \vec v_1 + t_2 \vec v_2 \implies \vec x = \vec {x_0} + t_1 \vec v_1 + t_2 \vec v_2\)

Finding a Vector Equation from Parametric Equations

\[x = 4 + 5t_1 - t_2\]
\[y = 2 - t_1 + 8t_2\]
\[z = t_1 + t_2\]

Solution

\[(x,y,z) = (4 + 5t_1 - t_2, 2 - t_1 + 8t_2, t_1 + t_2)\]
\[\implies (x,y,z) = (4,2,0) + (5t_1, -t_1, t_1) + (-t_2, 8t_2, t_2)\]
\[\implies (x,y,z) = (4,2,0) + t_1(5,-1,1) + t_2(-1,8,1)\]

Therefore, \(\vec {v_1} = (5, -1, 1)\) and \(\vec {v_2} = (-1, 8, 1)\).

References

Read more about notations and symbols.

  1. Read more about matrices. ↩↩↩↩

  2. Read more about vectors. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩

  3. Read more about sets. ↩↩↩↩↩

  4. Read more about planes. ↩↩

  5. Read more about lines. ↩↩↩