# What is Vector Triple Product

Here, you will learn what is vector triple product formula and linear independence and dependence of vectors.

Let’s begin –

## Vector Triple Product Formula

Let $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ be any three vectors, then the expression

$$\vec{a}\times (\vec{b}\times\vec{c})$$

is a vector & is called a vector triple product.

## Linear Independence And Dependence of Vectors

(a)  If $$\vec{x_1}$$, $$\vec{x_2}$$, ……….. $$\vec{x_n}$$ are n non zero vectors, & $$k_1$$, $$k_1$$, …… $$k_n$$ are n scalars & if the linear combination $$k_1\vec{x_1}$$ + $$k_2\vec{x_2}$$ + ….. $$k_n\vec{x_n}$$ = $$\vec{0}$$ $$\implies$$ $$k_1$$ = 0, $$k_2$$ = 0 ….. $$k_n$$ = 0 , then we say that vectors $$\vec{x_1}$$, $$\vec{x_2}$$, ……….. $$\vec{x_n}$$ are linearly independent vectors.

(b)  If $$\vec{x_1}$$, $$\vec{x_2}$$, ……….. $$\vec{x_n}$$ are not linearly independent then they are said to be linearly dependent vectors. i.e. if $$k_1\vec{x_1}$$ + $$k_2\vec{x_2}$$ + ….. $$k_n\vec{x_n}$$ = $$\vec{0}$$ & if there exists at least one $$k_r$$ $$\ne$$ 0 then $$\vec{x_1}$$, $$\vec{x_2}$$, ……….. $$\vec{x_n}$$ are said to be linearly dependent.

#### Fundamental Theorem in Space

Let $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ be non-zero, non-coplanar vectors in space. Then any vector $$\vec{r}$$, can be uniquely expressed as a linear combination of $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ i.e. There exist some unique x, y, z $$\in$$ R such that $$\vec{r}$$ = $$x\vec{a}$$ + $$y\vec{b}$$ + $$z\vec{c}$$

## Shortest Distance Between Two Lines

If two lines in space intersect at a point, then obviously the shortest distance between them is zero. Lines which do not intersect & also are not parallel are called skew lines. In other words the lines which are not coplanar are skew lines.

If two lines are given by   $$\vec{r_1}$$ = $$\vec{a_1}$$ + $$K_1\vec{b_1}$$ & $$\vec{r_2}$$ = $$\vec{a_2}$$ + $$K_2\vec{b_2}$$   then shortest distance between two lines are given by :

d = |$$(\vec{a_1} – \vec{a_1}).(\vec{b_1} \times \vec{b_2})\over |\vec{b_1} \times \vec{b_2}|$$|

If two lines are given by  $$\vec{r_1}$$ = $$\vec{a_1}$$ + $$K_1\vec{b}$$ & $$\vec{r_2}$$ = $$\vec{a_2}$$ + $$K_2\vec{b}$$ i.e. they are parallel, then shortest distance between two lines are given by :

d = |$$\vec{b} \times (\vec{a_1} – \vec{a_1})\over |\vec{b}|$$|