Ritika starts walking from his house to shopping mall. Instead of going to the mall directly, she first goes to a ATM, from there to her daughter's school and then reaches the mall. In the diagram, A, B, C and D represent the coordinates of House, ATM, School and Mall respectively.

Based on the above information, answer the following questions.
(i) Distance between House (A) and ATM (B) is

(ii) Distance between ATM (B) and School (C) is

(iii) Distance. between School (C) and Shopping mall (D) is

(iv) What is the total distance travelled by Ritika ?

(v) What is the extra distance travelled by Ritika in reaching the shopping mall?

Ginni purchased an air plant holder which is in the shape of a tetrahedron.
Let A, B, C and D are the coordinates of the air plant holder where A \(\equiv \) (1, 1, 1), B \(\equiv \) (2, 1, 3), C \(\equiv \) (3, 2, 2) and D \(\equiv \)(3, 3, 4).

Based on the above information, answer the following questions.
(i) Find the position vector of \(\overrightarrow{A B} \).

(ii) Find the position vector of \(\overrightarrow{A C} \).

(iii) Find the position vector of \(\overrightarrow{AD} .\).

(iv) Area of \(\Delta A B C\) = 

(v) Find the unit vector along \(\overrightarrow{AD} .\)

Geetika's house is situated at Shalimar Bagh at point 0, for going to Aloks house she first travels 8 km by bus in the East. Here at point A, a hospital is situated. From Hospital, Geetika takes an auto and goes 6 km in the North, here at point B school is situated. From school, she travels by bus to reach Aloks house which is at 30° East, 6 km from point B.

Based on the above information, answer the following questions.
(i) What is the vector distance between Geetikas house and school ?

(ii) How much distance Geetika travels to reach school?

(iii) What is the vector distance from school to Alok's house ?

(iv) What is the vector distance from Geetikas house to Alok's house?

(v) What is the total distance travelled by Geetika from her house to Alok's house?

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 ?

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

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

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

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

A plane started from airport situated at 0 with avelocity of 120 m/s towards east. Air is blowing at a velocity of 50 m/s towards the north as shown in the figure.
The plane travelled 1 hr in OP direction with the resultant velocity. From P to R the plane travelled 1 hr keeping velocity of 120 m/s and finally landed at R.

Based on the above information, answer the following questions.
(i) What is the resultant velocity from O to P?

(ii) What is the direction of travel of plane from O to P with East?

(iii) What is the displacement from O to P?

(iv) What is the resultant velocity from P to R?

(v) What is the displacement from P to R?

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

(ii) Position vector of D is

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

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

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

A building is to be constructed in the form of a triangular pyramid, ABCD as shown in the figure.

Let its angular points are A(0, 1, 2), B(3, 0, 1), C(4, 3, 6) and D(2, 3, 2) and G be the point of intersection of the medians of \(\Delta\)BCD.
Based on the above information, answer the following questions.
(i) The coordinates of point G are

(ii) The length of vector \(\overrightarrow{A G}\) is 

(iii) Area of \(\Delta\)ABC (in sq. units) is

(iv) The sum of lengths of \(\overrightarrow{A B}\) and \(\overrightarrow{A C}\) is

(v) The length of the perpendicular from the vertex D on the opposite face is

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 \( \vec{q} +\vec{r}\) =

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

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

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

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

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

(ii) Which of the following is not true?

(iii) Area of \(\Delta\)ABC is 

(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

(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 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

(ii) Position vector of B is

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

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

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