Students often turn to OP Malhotra ISC Class 12 Solutions Chapter 21 Vectors Ex 21(b) to clarify doubts and improve problem-solving skills.
S Chand Class 12 ICSE Maths Solutions Chapter 21 Vectors Ex 21(b)
Question 1.
\(\overrightarrow{O A}\) and \(\overrightarrow{O B}\) are vectors \(\vec{a}\) and \(\vec{b}\) respectively and X and Y are points of trisection of A B. Find, in terms of \(\vec{a}\) and \(\vec{b}\).
(i) \(\overrightarrow{\mathrm{OX}}\) and
(ii) \(\overrightarrow{\mathrm{OY}}\)
Answer:
Question 2.
\(\overrightarrow{O A}\) and \(\overrightarrow{O B}\) are vectors \(\vec{a}\) and \(\vec{b}\) respectively and P and Q are points \(\frac{1}{4}\) and \(\frac{3}{4}\) of the way along A B. Find, in terms of \(\vec{a}\) and \(\vec{b}\).
(i) \(\overrightarrow{\mathrm{OP}}\) and (ii) \(\overrightarrow{\mathrm{OQ}}\).
Answer:
Question 3.
A B C D is quadrilateral in which B C is paraliel to A D and the ratio of the lengths B C: A D is 4 : 7. Taking \(\overrightarrow{A B}\) and \(\overrightarrow{A D}\) as representatives of vectors \(\vec{v}\) and 7 \(\vec{u}\) respectively, find which vectors are represented by
(i) \(\overrightarrow{\mathrm{BC}}\)
(ii) \(\overrightarrow{\mathbf{A C}}\)
(iii) \(\overrightarrow{\mathrm{BD}}\)
(iv) \(\overrightarrow{\mathrm{DC}}\)
(v) \(\overrightarrow{\mathrm{AE}}\) where E is on BD such that B E = \(\frac{4}{11}\) BD in length;
(vi) \(\overrightarrow{\mathbf{A F}}\) where F is on AC such that AF = \(\frac{7}{11}\) AC.
Answer:
Question 4.
In fig. given below, B E is median of triangle A B C and G divides B E in the ratio 2 : 1.
(i) If \(\overrightarrow{\mathrm{AB}}\) represents \(\vec{u}\) and \(\overrightarrow{\mathrm{AC}}\) represents \(\vec{v}\), show that \(\overrightarrow{\mathrm{EB}}\) represents \(\vec{u}\) – \(\frac{1}{2}\) \(\vec{v}\) and \(\overrightarrow{A G}\) represents \(\frac{1}{3}\)(\(\vec{u}\) + \(\vec{v}\)).
(ii) If C F is a median, and H divides C F in the ratio 2 : 1, show that \(\overrightarrow{\mathrm{AH}}\) represents \(\frac{1}{3}\)(\(\vec{u}\) + \(\vec{v}\)).
(iii) If AD is a median and K divides AD in the ratio 2 : 1, which vector does \(\overrightarrow{A K}\) represents in terms of \(\overrightarrow{\boldsymbol{u}}\) and \(\overrightarrow{\boldsymbol{v}}\) ? What can you conclude about G, H, K ? What can you conclude about the medians of a triangle?
Answer:
Question 5.
Four points A, B, C, D with position vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\), \(\vec{d}\) respectively are
such that 3 \(\vec{a}\) – \(\vec{b}\) + 2 \(\vec{c}\) – 4 \(\vec{d}\) = \(\overrightarrow{0}\). Show that the four points are coplanar. Also, find the position vector of the points of intersection of lines AC and BD.
Answer:
This shows that the position vector of point P deviding AC in the ratio 2 : 3 is same as that of point dividing B D in the ratio 4 : 1. Hence A C and B D intersects at point P. Thus A, B, C and D are coplanar. Since P be the point of intersection of A C and B D.
Thus, P.V. of the point of intersection of lines AC and BD be \(\frac{3 \vec{a}+2 \vec{c}}{5}\) or \(\frac{\vec{b}+4 \vec{d}}{5}\).