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Case Study Questions

5 x 4 = 20

A barge is pulled into harbour by two tug boats as shown in the figure.

Based on the above information, answer the following questions.
(i) Position vector of A is

(a) \(4 \hat{i}+2 \hat{j}\)

(b) \(4 \hat{i}+10 \hat{j}\)

(c)\(4 \hat{i}-10 \hat{j}\)

(d) \(4 \hat{i}-2 \hat{j}\)

(ii) Position vector of B is

(a) \(4 \hat{i}+4 \hat{j}\)

(b) \(6 \hat{i}+6 \hat{j}\)

(c) \( 9 \hat{i}+7 \hat{j}\)

(d) \(3 \hat{i}+3 \hat{j}\)

(iii) Find the vector \(\vec{AC}\) in terms of \(\hat{i}, \hat{j}\)

(a) \(8 \hat{j}\)

(b) \(-8 \hat{j}\)

(c) \(8 \hat{i}\)

(d) None of these

(iv) If \(\vec{A}=\hat{i}+2 \hat{j}+3 \hat{k}\), then its unit vector is

(a)\(\frac{\hat{i}}{\sqrt{14}}+\frac{2 \hat{j}}{\sqrt{14}}+\frac{3 \hat{k}}{\sqrt{14}}\)

(b) \(\frac{3 \hat{i}}{\sqrt{14}}+\frac{2 \hat{j}}{\sqrt{14}}+\frac{\hat{k}}{\sqrt{14}}\)

(c) \(\frac{2 \hat{i}}{\sqrt{14}}+\frac{3 \hat{j}}{\sqrt{14}}+\frac{\hat{k}}{\sqrt{14}}\)

(d) None of these

(v) If \(\vec{A}=4 \hat{i}+3 \hat{j}\) and \(\vec{B}=3 \hat{i}+4 \hat{j}\), then IAI+ IBI = ___________.

(a) 12

(b) 13

(c) 14

(d) 10

Three slogans on chart papers are to be placed on a school bulletin board at the points A, Band C displaying A (Hub of Learning), B (Creating a better world for tomorrow) and C (Education comes first). The coordinates of these points are (1, 4, 2), (3, -3, -2) and (-2, 2, 6) respectively.

Based on the above information, answer the following questions.
(i) Let \(\vec{a}\), \(\vec{b}\)and \(\vec{c}\) be the position vectors of points A, B and C respectively, then \(\vec{a}\) + \(\vec{b}\)+ \(\vec{c}\) is equal to

(a) \(2 \hat{i}+3 \hat{j}+6 \hat{k} \)

(b) \(2 \hat{i}-3 \hat{j}-6 \hat{k}\)

(c) \(2 \hat{i}+8 \hat{j}+3 \hat{k} \)

(d) \(2(7 \hat{i}+8 \hat{j}+3 \hat{k})\)

(ii) Which of the following is not true?

(a) \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}=\overrightarrow{0}\)

(b) \(\overrightarrow{A B}+\overrightarrow{B C}-\overrightarrow{A C}=\overrightarrow{0}\)

(c) \(\overrightarrow{A B}+\overrightarrow{ BC}-\overrightarrow{C A}=\overrightarrow{0}\)

(d) \(\overrightarrow{A B}-\overrightarrow{C B}+\overrightarrow{C A}=\overrightarrow{0}\)

(iii) Area of \(\Delta\)ABC is

(a) 19 sq. units

(b) \(\sqrt 1937 sq. unit\)

(c) \(\frac{1}{2}\sqrt 1937 sq. unit\)

(d) \(\sqrt 1837 sq. unit\)

(iv) Suppose, if the given slogans are to be placed on a straight line, then the value of \(|\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|\) will be equal to

(a) -1

(b) -2

(c) 2

(d) 0

(v) If \(\vec{a}=2 \hat{i}+3 \hat{j}+6 \hat{k}\) then unit vector in the direction of vector \(\vec{a}\)is

(a) \(\frac{2}{7} \hat{i}-\frac{3}{7} \hat{j}-\frac{6}{7} \hat{k}\)

(b) \(\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}+\frac{6}{7} \hat{k}\)

(c) \(\frac{3}{7} \hat{i}+\frac{2}{7} \hat{j}+\frac{6}{7} \hat{k}\)

(d) None of these

If two vectors are represented by the two sides of a triangle taken in order, then their sum is represented by the third side of the triangle taken in opposite order and this is known as triangle law of vector addition. lased on the above information, answer the following questions.
(i) If \(\vec{p}, \vec{q}, \vec{r}\)are the vectors represented by the sides of a triangle taken in order, then \( \vec{q} +\vec{r}\) =

(a) \(\vec{p}\)

(b) \(2 \vec{p}\)

(c) \(-\vec{p}\)

(d) None of these

(ii) If ABCD is a parallelogram and AC and BD are its diagonals, then\( \vec{AC} +\vec{BD}\) =

(a) \(2 \vec{DA}\)

(b) \(2 \vec{AB}\)

(c) \(2\overrightarrow{BC}\)

(d) \(2\vec{BD}\)

(iii) If ABCD is a parallelogram, where \(\overrightarrow{A B}\)\(=2\overrightarrow{a}\) and \(\overrightarrow{BC}\) \(=2\overrightarrow{b}\), then \( \vec{AC} -\vec{BD}\) =

(a) \(3\vec{a}\)

(b) \(4\vec{a}\)

(c) \(2\vec{b}\)

(d) \(4\vec{b}\)

(iv) If ABCD is a quadrilateral whose diagonals are \( \vec{AC}\) and \(\vec{BD}\), then \( \vec{BA} +\vec{CD}\) =

(a) \(\overrightarrow{A C}+\overrightarrow{D B}\)

(b) \(\overrightarrow{A C}+\overrightarrow{B D}\)

(c) \(\overrightarrow{B C}+\overrightarrow{A D}\)

(d) \(\overrightarrow{B D}+\overrightarrow{C A}\)

(v) If T is the mid point of side YZ of \(\triangle\)XYZ, then\(\overrightarrow{XY}\) + \(\overrightarrow{XZ}\) =

(a) \(2\vec{YT}\)

(b) \(2\vec{XT}\)

(c) \(2\vec{TZ}\)

(d) None of these

Ishaan left from his village on weekend. First, he travelled up to temple. After this, he left for the zoo. After this he left for shopping in a mall. The positions of Ishaan at different places is given in the following graph.

Based on the above information, answer the following questions.
(i) Position vector of B is

(a) \(3 \hat{i}+5 \hat{j}\)

(b) \(5 \hat{i}+3 \hat{j}\)

(c) \(-5 \hat{i}-3 \hat{j}\)

(d) \(-5 \hat{i}+3 \hat{j}\)

(ii) Position vector of D is

(a) \(5 \hat{i}+3 \hat{j}\)

(b) \(3 \hat{i}+5 \hat{j}\)

(c) \(8 \hat{i}+9 \hat{j}\)

(d) \(9 \hat{i}+8 \hat{j}\)

(iii) Find the vector \(\overrightarrow{B C}\) in terms of \(\hat{i}, \hat{j}\).

(a) \( \hat{i}-2 \hat{j}\)

(b) \( \hat{i}+2 \hat{j}\)

(c) \(2\hat{i}+ \hat{j}\)

(d) \(2\hat{i}- \hat{j}\)

(iv) Length of vector \(\overrightarrow{A D}\) is

(a) \(\sqrt 67\) units

(b) \(\sqrt 85\) units

(c) 90 units

(d) 100 units

(v) If \(\vec{M}=4 \hat{\jmath}+3 \hat{k}\) , then its unit vector

(a) \(\frac{4}{5} \hat{j}+\frac{3}{5} \hat{k}\)

(b) \(\frac{4}{5} \hat{j}-\frac{3}{5} \hat{k}\)

(c) \(-\frac{4}{5} \hat{j}+\frac{3}{5}\hat{k}\)

(d) \(-\frac{4}{5} \hat{j}-\frac{3}{5} \hat{k}\)

Teams A, B, C went for playing a tug of war game. Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area (team areas shown below).
Team A pulls with force F₁ = \(\hat{4}+\hat{0} \hat{j}\) KN
Team B ⟶ F₂ = \(-2 \hat{i}+4 \hat{j}\) KN
Team C ⟶ F₃ = \(-3 \hat{i}-3 \hat{j}\) KN

Based on the above information, answer the following questions.
(i) Which team will win the game ?

(a) Team B

(b) Team A

(c) Team C

(d) No one

(ii) What is the magnitude of the teams combined force ?

(a) 7 KN

(b) 1.4 KN

(c) 1.5 KN

(d) 2 KN

(iii) In what direction is the ring getting pulled?

(a) 2.0 radian

(b) 2.5 radian

(c) 2.4 radian

(d) 3 radian

(iv) What is the magnitude of the force of Team B?

(a) 2\(\sqrt 5\) KN

(b) 6 KN

(c) 2 KN

(d) \(\sqrt 6\) KN

(v) How many KN force is applied by Team A?

(a) 5 KN

(b) 4 KN

(c) 2 KN

(d) 16 KN

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Answers Key:

Case Study Questions

5 x 4 = 20

(i) (b): Here, (4, 10) are the coordinates of A.
\(\therefore\) P.V. of A = \(4 \hat{i}+10 \hat{j}\)
(ii) (c): Here, (9, 7) are the coordinates of B.
\(\therefore\) P.V. of B \(=9 \hat{i}+7 \hat{j}\)
(iii) (b): Here, P.V. of A = \(4 \hat{i}+10 \hat{j}\) and P.V. of \(C=4 \hat{i}+2 \hat{j}\)
\(\therefore \ \overrightarrow{A C}=(4-4) \hat{i}+(2-10) \hat{j}=-8 \hat{j}\)
(iv) (a): Here \(\vec{A}=\hat{i}+2 \hat{j}+3 \hat{k}\)
\( \therefore \ |\vec{A}|=\sqrt{1^{2}+2^{2}+3^{2}}=\sqrt{1+4+9}=\sqrt{14} \)
\(\therefore \ \hat{A}=\frac{\vec{A}}{|\vec{A}|}=\frac{\hat{i}+2 \hat{j}+3 \hat{k}}{\sqrt{14}}=\frac{1}{\sqrt{14}} \hat{i}+\frac{2}{\sqrt{14}} \hat{j}+\frac{3}{\sqrt{14}} \hat{k}\)
(v) (d): We have, \(\vec{A}=4 \hat{i}+3 \hat{j}\) and \( \vec{B}=3 \hat{i}+4 \hat{j}\)
\(\therefore|\vec{A}|=\sqrt{4^{2}+3^{2}}=\sqrt{16+9}=\sqrt{25}=5\)
and \(|\vec{B}|=\sqrt{3^{2}+4^{2}}=\sqrt{9+16}=\sqrt{25}=5\)
Thus, \(|\vec{A}|+|\vec{B}|=5+5=10\)

(i) (a) : \(\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-3 \hat{j}-2 \hat{k}\) and \(\vec{c}=-2 \hat{i}+2 \hat{j}+6 \hat{k}\)
\(\therefore \ \vec{a}+\vec{b}+\vec{c}=2 \hat{i}+3 \hat{j}+6 \hat{k}\)
(ii) (c): Using triangle law of addition in \(\Delta\)ABC, we get
\(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}=\overrightarrow{0}\) which can be rewritten as
\(\overrightarrow{A B}+\overrightarrow{B C}-\overrightarrow{A C}=\overrightarrow{0} \text { or } \overrightarrow{A B}-\overrightarrow{C B}+\overrightarrow{C A}=\overrightarrow{0}\)
(iii) (c) : We have, A(1, 4, 2), B(3, -3, -2) and C(-2, 2, 6)
Now, \(\overrightarrow{A B}=\vec{b}-\vec{a}=2 \hat{i}-7 \hat{j}-4 \hat{k}\)
and \(\overrightarrow{A C}=\vec{c}-\vec{a}=-3 \hat{i}-2 \hat{j}+4 \hat{k}\)
\(\therefore \overrightarrow{A B} \times \overrightarrow{A C}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & -7 & -4 \\ -3 & -2 & 4 \end{array}\right|\)
\(=\hat{i}(-28-8)-\hat{j}(8-12)+\hat{k}(-4-21)=-36 \hat{i}+4 \hat{j}-25 \hat{k}\)
Now, \(|\overrightarrow{A B} \times \overrightarrow{A C}|=\sqrt{(-36)^{2}+4^{2}+(-25)^{2}}\)
\(=\sqrt{1296+16+625}=\sqrt{1937}\)
\(\therefore \text { Area of } \Delta A B C=\frac{1}{2}|\overrightarrow{A B} \times \overrightarrow{A C}|=\frac{1}{2} \sqrt{1937} \text { sq. units }\)
(iv) (d): If the given points lie on the straight line, then the points will be collinear and so area of \( \Delta A B C=0 \)
\(\Rightarrow |\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|=0\)
If a, b, c are the position vectors of the three vertices A, Band C of \( \Delta A B C \), then area of triangle
\(\left.=\frac{1}{2}|\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|\right]\)
(v) (b): Here, \( |\vec{a}| =\sqrt{2^{2}+3^{2}+6^{2}}=\sqrt{4+9+36} \)
\(=\sqrt{49}=7\)
\(\therefore\) Unit vector in the direction of vector \(\vec{a}\) is
\(\hat{a}=\frac{2 \hat{i}+3 \hat{j}+6 \hat{k}}{7}=\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}+\frac{6}{7} \hat{k}\)

(i) (c) : Let OAB be a triangle such that

\(\overrightarrow{A O}=-\vec{p}, \overrightarrow{A B}=\vec{q}, \)\(\overrightarrow{B O}=\vec{r}\)
Now, \(\vec{q}+\vec{r} =\overrightarrow{A B}+\overrightarrow{B O}\)
\(=\overrightarrow{A O} =-\vec{p}\)
(ii) (c) : From triangle law of vector addition,
\(\overrightarrow{A C}+\overrightarrow{B D}=\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{B C}+\overrightarrow{C D}\)

\( =\overrightarrow{A B}+2 \overrightarrow{B C}+\overrightarrow{C D}\)
\(=\overrightarrow{A B}+2 \overrightarrow{B C}-\overrightarrow{A B}=2 \overrightarrow{B C} \)
(iii) (b) : \(\operatorname{In} \Delta A B C, \overrightarrow{A C}=2 \vec{a}+2 \vec{b}\)

and in \(\Delta A B D, 2 \vec{b}=2 \vec{a}+\overrightarrow{B D}\)..(ii)
[By triangle law of addition]
Adding (i) and (ii), we have
\( \overrightarrow{A C}+2 \vec{b}=4 \vec{a}+\overrightarrow{B D}+2 \vec{b} \)
⇒\( \overrightarrow{A C}-\overrightarrow{B D}=4 \vec{a} \)
(iv) (d) : \(\text { In } \Delta A B C, \overrightarrow{B A}+\overrightarrow{A C}=\overrightarrow{B C}\)... (i)
[By triangle law]
In \(\Delta B C D, \overrightarrow{B C}+\overrightarrow{C D}=\overrightarrow{B D}\) ..(i)
From (i) and (ii), \(\overrightarrow{B A}+\overrightarrow{A C}=\overrightarrow{B D}-\overrightarrow{C D}\)
\(\Rightarrow \overrightarrow{B A}+\overrightarrow{C D}=\overrightarrow{B D}-\overrightarrow{A C}=\overrightarrow{B D}+\overrightarrow{C A}\)
(v) (b): Since T is the mid point of YZ.
So,\(\overrightarrow{Y T}=\overrightarrow{T Z}\)
Now,\(\overrightarrow{X Y}+\overrightarrow{X Z}=(\overrightarrow{X T}+\overrightarrow{T Y})+(\overrightarrow{X T}+\overrightarrow{T Z})\)
[By triangle law]
\(=2 \overrightarrow{X T}+\overrightarrow{T Y}+\overrightarrow{T Z}=2 \overrightarrow{X T} \quad[\because \overrightarrow{T Y}=-\overrightarrow{Y T}]\)

(i) (b): Here (5, 3) are the coordinates of B.
\(\therefore\) P.V. of B = \(5 \hat{i}+3 \hat{j}\)
(ii) (d) : Here (9, 8) are the coordinates of D.
\(\therefore\) P.V. of D = \({9} \hat{i}+8 \hat{j}\)
(iii) (b) : P.V. of B = \(5 \hat{i}+3 \hat{j}\) and ஃ P.V. of C = \(6 \hat{i}+5 \hat{j}\)
\(\therefore \ \overrightarrow{B C}=(6-5) \hat{i}+(5-3) \hat{j}=\hat{i}+2 \hat{j}\)
(iv) (b): Since P.V. of A = \(2 \hat{i}+2 \hat{j}\) , P.V. of D = \({9} \hat{i}+8 \hat{j}\)
\( \therefore \overrightarrow{A D}=(9-2) \hat{i}+(8-2) \hat{j}=7 \hat{i}+6 \hat{j} \)
\(|\overrightarrow{A D}|^{2}=7^{2}+6^{2}=49+36=85 \)
\(\Rightarrow |\overrightarrow{A D}|=\sqrt{85} \text { units }\)
(v) (a): We have, \(\vec{M}=4 \hat{j}+3 \hat{k}\) ,
\( \therefore |\vec{M}|=\sqrt{4^{2}+3^{2}}=\sqrt{16+9}=\sqrt{25}=5 \)
\(\therefore \hat{M}=\frac{\vec{M}}{|\vec{M}|}=\frac{4 \hat{j}+3 \hat{k}}{5}=\frac{4}{5} \hat{j}+\frac{3}{5} \hat{k}\)

Here, \(\left|\vec{F}_{1}\right|=\sqrt{(4)^{2}+0^{2}}=4 \mathrm{KN}\)
\( \left|\vec{F}_{2}\right|=\sqrt{(-2)^{2}+4^{2}}=\sqrt{20} \mathrm{KN} \)
\(\left|\vec{F}_{3}\right|=\sqrt{(-3)^{2}+(-3)^{2}}=\sqrt{18} \mathrm{KN}\)
(i) (a): Since, \(\sqrt 20\) is larger. So, team B will win the game.
(ii) (b): Let F be the combined force
\(\therefore \vec{F}=\vec{F}_{1}+\vec{F}_{2}+\vec{F}_{3}=4 \hat{i}+0 \hat{j}-3 \hat{i}-3 \hat{j}-2 \hat{i}+4 \hat{j} \)
\(=-\hat{i}+\hat{j} \)
\(\therefore |\vec{F}|=\sqrt{(-1)^{2}+1^{2}}=\sqrt{2}=1.4 \mathrm{KN}\)
(iii) (c) : We have, \(\vec{F}=-\hat{i}+\hat{j}\)
\(\therefore \theta=\tan ^{-1}\left(\frac{F_{y}}{F_{x}}\right)=\tan ^{-1}\left(\frac{1}{-1}\right)=\frac{3 \pi}{4} \text { radian }\)
= 0.75 x 3.14 radian = 2.3555 radian ≈ 2.4 radian
(iv) (a): Magnitude of force of Team B = \(\sqrt 20\) KN
= 2\(\sqrt 5\) KN
(v) (b): 4 KN force is applied by team A.