Examples On Vector Cross Product Set-2

Example – 25

Let A_i, i = 1, 2, 3, 4 be the areas of the faces of a tetrahedron. Let \({\vec n_i}\), i = 1, 2, 3, 4 be the outward drawn normals to the respective faces with magnitudes equal to the corresponding areas.

Prove that \({\vec n_1} + {\vec n_2} + {\vec n_3} + {\vec n_4} = \vec 0\)

Solution: There is no loss of generality in assuming one vertex A to be the origin \(\vec 0\) . Let the other vertices be  \(B(\vec b),\;C(\vec c)\;{\text{and}}\;D(\vec d)\) .

In the following, note carefully how the order of the cross-product is taken:

We have

\[\begin{align}&{{\vec n}_1} = \frac{1}{2}(\overrightarrow {BC}  \times \overrightarrow {BD} ) \hfill \\\\&\quad= \frac{1}{2}\left\{ {(\vec c - \vec b) \times (\vec d - \vec b)} \right\} \hfill \\\\&\quad= \frac{1}{2}\left\{ {\vec c \times \vec d + \vec b \times \vec c + \vec d \times \vec b} \right\} \hfill \\\\&  {{\vec n}_2} = \frac{1}{2}(\overrightarrow {AD}  \times \overrightarrow {AC} ) \hfill \\\\&\quad  = \frac{1}{2}(\vec d \times \vec c) =  - \frac{1}{2}(\vec c \times \vec d) \hfill \\\\&
  {{\vec n}_3} = \frac{1}{2}(\overrightarrow {AB}  \times \overrightarrow {AD} ) \hfill \\\\&\quad = \frac{1}{2}(\vec b \times \vec d) =  - \frac{1}{2}(\vec d \times \vec b) \hfill\\ \\&  {{\vec n}_4} = \frac{1}{2}(\overrightarrow {AC}  \times \overrightarrow {AB} ) \hfill \\\\&\quad   = \frac{1}{2}(\vec c \times \vec b) =  - \frac{1}{2}(\vec b \times \vec c) \hfill \\ \end{align} \]

From these relations, it is clear that

\[{\vec n_1} + {\vec n_2} + {\vec n_3} + {\vec n_4} = \vec 0\]

Example – 26

Let \(\vec a = 2\hat i + \hat k,\;\;\vec b = \hat i + \hat j + \hat k\;\;{\text{and}}\;\;\vec c = 4\hat i - 3\hat j + 7\hat k\). Find a vector \(\vec r\)  such that \(\vec r \times \vec b = \vec c \times \vec b\;{\text{and}}\;\vec r \cdot \vec a = 0\)

Solution:                                                                                                                 \(\vec r \times \vec b = \vec c \times \vec b \)

\[\begin{align} & \Rightarrow \quad  (\vec r - \vec c) \times \vec b = \vec 0 \hfill \\\\&\Rightarrow  \quad \vec r - \vec c\,is\,{\text{ }}parallel\,{\text{ }}to\,\vec b \hfill \\ \end{align} \]

Thus,

\[\begin{align}&\vec r - \vec c = \lambda \vec b\,\,for{\text{ }}some\,\,\lambda  \in \mathbb{R} \hfill \\\\& \Rightarrow \quad   \vec r = \vec c + \lambda \vec b\quad\qquad\qquad....\left( 1 \right) \hfill \\ \end{align} \]

Now, since

\[\begin{align}&\qquad\vec r \cdot \vec a = 0 \hfill \\\\&  \Rightarrow   \quad(\vec c + \lambda \vec b) \cdot \vec a = 0 \hfill \\\\&\Rightarrow  \quad \vec a \cdot \vec c + \lambda \vec a \cdot \vec b = 0 \hfill \\\\&
   \Rightarrow \quad   \lambda  =  - \frac{{\vec a \cdot \vec c}}{{\vec a \cdot \vec b}} \hfill \\\\&\qquad\quad=  - \frac{{8 + 7}}{{2 + 0 + 1}} \hfill \\\\&\quad\qquad  = {\text{ }}-{\text{ }}5 \hfill \\ 
\end{align} \]

Using (1), we can now determine  \(\vec r\) :

\[\begin{align}&\;\vec r = \vec c + \lambda \vec b \hfill \\\\&\;\;= \vec c - 5\vec b \hfill \\\\&\;\; = (4\hat i - 3\hat j + 7\hat k) - 5(\hat i + \hat j + \hat k) \hfill \\\\&\;\;=  - \hat i - 8\hat j + 2\hat k \hfill \\ \end{align} \]

TRY YOURSELF - III

Q. 1 If A, B, C, D be any four points in space, prove that

\[\left| {\overrightarrow {AB}  \times \overrightarrow {CD}  + \overrightarrow {BC}  \times \overrightarrow {AD}  + \overrightarrow {CA}  \times \overrightarrow {BD} } \right| = 4 \times \;{\text{Area}}\;(\Delta ABC)\]

Q. 2 Let \(\overrightarrow {OA}  = \vec a,\;\;\overrightarrow {OB}  = 10\vec a + 2\vec b\;{\text{and}}\;\overrightarrow {OC}  = \vec b\) , where O is the origin. Let p denote the area of the quadrilateral OABC and q denote the area of the parallelogram with OA and OC as adjacent sides. Prove that p = 6q.

Q. 3 If A₁, A₂, A₃, ... A_n   are the vertices of a regular plane polygon with n sides and O is its centre, prove that

\[\sum\limits_{i = 1}^{n - 1} {} ({\overrightarrow {OA} _i} \times {\overrightarrow {OA} _{i + 1}}) = (n - 1)\;({\overrightarrow {OA} _1} \times {\overrightarrow {OA} _2})\]

Q. 4 Let  \(\vec a = \hat i + \hat j + \hat k\;\,{\text{and}}\;\,\vec b = \hat j - \hat k\) . Find a vector \(\vec r\) such that  \(\vec a \times \vec r = \vec b\) and \(\vec a \cdot \vec r = 3\)

Q. 5 Prove the sine rule for a triangle ABC with sides a, b, c :

\[\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}}\]