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