Examples On Vector Dot Product Set-3

Example – 21

If a, b, c are the lengths of the sides of  \(\Delta ABC\) opposite to the angles A, B and C respectively, prove using vector methods that

\[a(1 + \cos A) + b(1 + \cos B) + c(1 + \cos C) = (a + b + c)(\cos A + \cos B + \cos C)\]

Solutions:

We have, by the triangle law,

\[\begin{align}&\qquad\;\;\vec a + \vec b + \vec c = \vec 0 \hfill \\\\ &\Rightarrow \quad \vec a =  - (\vec b + \vec c) \hfill \\\\ &\Rightarrow \quad   \vec a \cdot \vec a =  - \vec a \cdot (\vec b + \vec c) \hfill \\\\& \Rightarrow \quad {a^2} = ab\cos C + ac\cos B\left( \begin{gathered} \because \vec a \cdot \vec b =  - ab\cos C \hfill \\\\\,\,\,\,\,\vec a \cdot \vec c =  - ac\cos B \hfill \\\\ \end{gathered}  \right) \hfill \\\\& \Rightarrow \quad  a = b\cos C + c\cos B\qquad\qquad\qquad...\left( 1 \right) \hfill \\ 
\end{align} \]

Similarly,

\[\begin{align}&b = c\cos A + a\cos C\qquad\qquad\qquad...\left( 2 \right) \hfill \\\\& c = a\cos B + b\cos A\qquad\qquad\qquad...\left( 3 \right) \hfill \\ \end{align} \]

Adding (1), (2) and (3), we have

\[a + b + c = a(\cos B + \cos C) + b(\cos C + \cos A) + c(\cos A + \cos B)\]

Adding \(a\cos A + b\cos B + c\cos C\) on both sides, we have

\[a(1 + \cos A) + b(1 + \cos B) + c(1 + \cos C) = (a + b + c)(\cos A + \cos B + \cos C)\]

Example – 22

Find three-dimensional vectors \({\vec v_1},\;\;{\vec v_2}\;\;{\text{and}}\;\;{\vec v_3}\) satisfying the relations

\[\begin{align}& {{\vec v}_1} \cdot {{\vec v}_1} = 4 && {{\vec v}_1} \cdot {{\vec v}_2} =  - 2 && {{\vec v}_1} \cdot {{\vec v}_3} = 6 \hfill \\\\&{{\vec v}_2} \cdot {{\vec v}_2} = 2 && {{\vec v}_2} \cdot {{\vec v}_3} =  - 5 & &{{\vec v}_3} \cdot {{\vec v}_3} = 29 \hfill \\ \end{align} \]

Solutions: A reference frame for the vectors has not been specified; therefore, it is up to us to choose a reference frame and then use it consistently and evaluate the required vectors in that reference frame.

Assume \({\vec v_1}\) to be along the x-direction, i.e.

\[\begin{align}&\qquad\;  {{\vec v}_1} = 2\hat i \hfill \\\\& Let\quad{{\vec v}_2} = a\hat i + b\hat j + c\hat k \hfill \\\\&\qquad\;\; {{\vec v}_3} = p\hat i + q\hat j + r\hat k \hfill \\ \end{align} \]

Now we step by step use all the given relations to determine the unknown constraints:

\[\begin{align}& {{\vec v}_1} \cdot {{\vec v}_2} =  - 2 \quad  \Rightarrow  \quad 2a =  - 2 \hfill \\\\&\qquad\qquad\qquad \Rightarrow\quad   a =  - 1\quad\qquad\qquad\;...\left( 1 \right) \hfill \\\\&
  {{\vec v}_2} \cdot {{\vec v}_2} = 2 \quad\;\;  \Rightarrow\quad   {a^2} + {b^2} + {c^2} = 2 \hfill \\\\&
 \qquad\qquad\qquad  \Rightarrow \quad   {b^2} + {c^2} = 1\quad\qquad\quad...\left( 2 \right){\text{ }}\left( {from{\text{ }}\left( 1 \right)} \right) \hfill \\\\&{{\vec v}_1} \cdot {{\vec v}_3} = 6 \quad \;\;  \Rightarrow \quad  2p = 6 \hfill \\\\ &\qquad\qquad\qquad \Rightarrow \quad p = 3\qquad\qquad\qquad\;\;...\left( 3 \right) \hfill \\\\& {{\vec v}_2} \cdot {{\vec v}_3} =  - 5  \quad \Rightarrow  \quad ap + bq + cr =  - 5 \hfill \\\\&\qquad\qquad\qquad\;\Rightarrow  \quad bq + cr =  - 2\qquad\;\;\;\;\;...\left( 4 \right){\text{ }}\left( {using{\text{ }}\left( 1 \right){\text{ }}and{\text{ }}\left( 3 \right)} \right) \hfill \\\\&\;\;  {{\vec v}_3} \cdot {{\vec v}_3} = 29 \quad \Rightarrow  \quad {p^2} + {q^2} + {r^2} = 29 \hfill \\\\&\qquad\qquad\qquad\;\Rightarrow \quad  {q^2} + {r^2} = 20\qquad\;\;\;\;\;\;...\left( 5 \right){\text{ }}\left( {using{\text{ }}\left( 3 \right)} \right) \hfill \\ \end{align} \]

Notice that (2), (4) and (5) are three equations in four unknowns. To get over this problem (it is not a problem actually! There will be an infinite set of vectors satisfying the given constraints. We have to find any one of them), when we chose \({{\vec v}_1}\) to be along the x-axis, we could also have adjusted the co-ordinate frame, so that \({\vec v_1}\;\;{\text{and}}\;\;{\vec v_2}\) lie in the x – z plane. This can always be done; since it is upto us to choose the frame of reference, we chose it so that the x – z plane co-insides with the plane of \({\vec v_1}\;\;{\text{and}}\;\;{\vec v_2}\).

How does this help? Now we’ll have one unknown less, since the y-component of \({{\vec v}_2}\) is zero, i.e., b = 0.

Thus, (2), (4) and (5) reduce to

\[{c^2} = 1, \,\,\,\,\,\,cr = 0 - 2,  \,\,\,\,\,\,{q^2} + {r^2} = 20\]

\[ \Rightarrow \quad c = \pm \;1,\,\,\,\,\,\,\,\,r = \mp \;2, \,\,\,\,\,\,\,\,q = \pm 4\]

Thus, the three dimensional vectors that satisfy the given constraints can be

\[{\vec v_1} = 2\hat i  \,\,\,\,\,\,\,\,\,{\vec v_2} = - \hat i + \hat k  \,\,\,\,\,\,\,\,\, {\vec v_3} = 3\hat i \pm 4\hat j - 2\hat k\]

OR

\[{\vec v_1} = 2\hat i \,\,\,\,\,\,\,\,\,{\vec v_2} = - \hat i - \hat k  \,\,\,\,\,\,\,\,\,{\vec v_3} = 3\hat i \pm 4\hat j + 2\hat k\]

To emphasize once again, we were required to find vectors satisfying the given constraints. This meant that absolute positions of the vectors were not important; what mattered was their relative sizes and orientation; and thus the coordinate axes was our choice. We selected it in a way which made the calculations most convenient.

TRY YOURSELF - II

Q. 1 Determine the values of c possible so that for all real x, the vectors \(cx\hat i - 6\hat j + 3\hat k\;and\;x\hat i + 2\hat j + 2cx\hat k\) and make an obtuse angle with each other.

Q. 2 Constant forces \({F_1} \equiv (2\hat i - 5\hat j + 6\hat k)N\;and\;{\vec F_2} \equiv ( - \hat i + 2\hat j - \hat k)N\) act on a particle and the particle is displaced from \(A \equiv (4\hat i - 3\hat j - 2\hat k)m\;to\;B \equiv (6\hat i + \hat j - 3\hat k)m\) .

Find the total work done by the forces.

Q. 3 Show that the diagonals of a rhombus bisect each other at right angles.

Q. 4 Using vectors, prove the trigonometric relation

\[\cos (A \pm B) = \cos A\cos B \mp \sin A\sin B\]

Q. 5 Prove that the perpendicular bisectors of the sides of a triangle are concurrent.

Q. 6 In \(\Delta ABC\)  with sides a, b, c opposite to angles A, B, C respectively, prove that

(i) \({a^2} = {b^2} + {c^2} - 2bc\;\;\cos A\)

(ii) \(\begin{align}a\cos B - b\cos A = \frac{{{a^2} - {b^2}}}{c}\end{align}\)

Q. 7 Find the unit vector which makes equal angles with the vectors \((\hat i - 2\hat j + 2\hat k),\,\,( - 4\hat i - 3\hat k)\;\;{\text{and}}\;\;\hat j\)

Q. 8 The lengths of the sides a, b, c of \(\Delta ABC\)  (a, b, c opposite to A, B, C respectively) satisfy the relation \({a^2} + {b^2} = 5{c^2}\) . Prove that the medians drawn to the sides with lengths a and b, are perpendicular.

Q. 9 Find the possible values of a for which the vector \(\vec r = ({a^2} - 4)\,\hat i + 2\hat j - ({a^2} - 9)\hat k\) makes acute angles with the coordinate axes.

Q. 10 Prove using vector methods that the angle in a semi-circle is a right angle.