Related InterviewSolutions

• Write the length (magnitude) of a vector whose projections on the coordinate axes are 12, 3 and 4 units.
• Suppose the floor of a hotel is made up of mirror polished Salvatore stone. There is a large crystal chandelier attached to the ceiling of the hotel room. Consider the floor of the hotel room as a plane having the equation x – y + z = 4 and the crystal chandelier is suspended at the point (1, 0, 1).Based on the above answer the following:1. Find the direction ratios of the perpendicular from the point (1, 0, 1) to the plane x – y + z = 4.a. (-1, -1, 1)b. (1, -1, -1)c. (-1, -1, -1)d. (1, -1, 1)2. Find the length of the perpendicular from the point (1, 0, 1) to the plane x – y + z = 4.a. \(\cfrac2{\sqrt3}\) unitsb. \(\cfrac4{\sqrt3}\) unitsc. \(\cfrac6{\sqrt3}\) unitsd. \(\cfrac8{\sqrt3}\)units3. The equation of the perpendicular from the point (1, 0, 1) to the plane x – y + z = 4 is4. The equation of the plane parallel to the plane x – y + z = 4, which is at a unit distance from the point (1, 0, 1) isa. x – y + z + (2 - √3 )b. x – y + z - (2 + √3 )c. x – y + z + (2 + √3 )d. Both (a) and (c)5. The direction cosine of the normal to the plane x – y + z = 4 is
• A mobile tower stands at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, -1, 1) and C(1, 2, 1) on it. The mobile tower is tied with 3 cables from the point A, B and C such that it stands vertically on the ground. The top of the tower is at the point (2, 3, 1) as shown in the figure.Based on the above answer the following:1. The equation of the plane passing through the points A, B and C isa. 3x - 2y + 4z = -11b. 3x + 2y + 4z = 11c. 3x - 2y - 4z = 11d. -3x + 2y + 4z = -112. The height of the tower from the ground isa. \(\cfrac5{\sqrt29}\) unitsb. \(\cfrac7{\sqrt29}\) unitsc. \(\cfrac6{\sqrt29}\) unitsd. \(\cfrac8{\sqrt29}\) units3. The equation of the perpendicular line drawn from the top of the tower to the ground is 4. The coordinates of the foot of the perpendicular drawn from the top of the tower to the ground are5. The area of ∆ isa. \(\cfrac{\sqrt{29}}4\) sq. unitsb. \(\cfrac{\sqrt{29}}2\) sq. unitsc. \(\cfrac{\sqrt{39}}2\) sq. unitsd. \(\cfrac{\sqrt{39}}4\) sq. units
• Show that the points A(-2i + 3j + 5k), B(i + 2j + 3k), C(7i - k) are collinear.
• If \(\bar a\) = 3i + j + 2k,(i) Find the magnitude of \(\bar a\).(ii) If the projection of \(\bar a\) on another vector \(\bar b\) is \(\sqrt{14}\), which among the following could be \(\bar b\) ?(a) i + j + k(b) 6i + 2j + 4k(c) 3i – j + 2k(d) 2i + 3j + k(iii) If \(\bar a\) makes an angle 60° with a vector \(\bar c\), find the projection of \(\bar a\) on \(\bar c\)
• Write two different vectors having same magnitude.
• Solar Panels have to be installed carefully so that the tilt of the roof, and the direction to the sun, produce the largest possible electrical power in the solar panels.A surveyor uses his instrument to determine the coordinates of the four corners of a roof where solar panels are to be mounted. In the picture , suppose the points are labelled counter clockwise from the roof corner nearest to the camera in units of meters P1 (6, 8, 4) , P2 (21, 8, 4), P3 (21,16,10) and P4 (6,16,10)1. What are the components to the two edge vectors defined by \(\vec A\)= PV of P2 – PV of P1 and \(\vec B\) = PV of P4 – PV of P1? (where PV stands for position vector)2. Write the vector in standard notation with \(\hat i,\hat j\) and \(\hat k\) (where \(\hat i,\hat j\) and \(\hat k\) are the unit vectors along the three axes).3. What are the magnitudes of the vectors \(\vec A\) and \(\vec B\)and in what units?4. What are the components to the vector \(\vec N\), perpendicular to \(\vec A\) and \(\vec B\) and the surface of the roof?5. What is the magnitude of \(\vec N\)and its units? The sun is located along the unit vector\(\vec S=1/2 \,\hat i- 6/7\, \hat j+1/7\,\hat k\). If the flow of solar energy is given by the vector \(\vec F\)= 910 S in units of watts/meter2 , what is the dot product of vectors \(\vec F\) with \(\vec N\), and the units for this quantity?6. What is the angle between vectors \(\vec N\) and \(\vec S\)? What is the elevation angle of the sun above the plane of the roof? (COS 510°=0.629 )
• A class XII student appearing for a competitive examination was asked to attempt the following questions.Let  \(\vec a,\vec b\) and \(\vec c\) be three non zero vectors.1. If  \(\vec a\) and  \(\vec b\) are such that \(|\vec a+\vec b|=|\vec a-\vec b|\) thena. \(\vec a\perp\vec b\)b. \(\vec a||\vec b\) c. \(\vec a=\vec b\)d. None of these2. If \(\vec a=\hat i-2\hat j,\) \(\vec b=2\hat i+\hat j+3\hat k\) then evaluate \((2\vec a+\vec b).[(2\vec a+\vec b)\times(\vec a-2\vec b)]\)a. 0b. 4c. 3d. 23. If  \(\vec a\) and  \(\vec b\)are unit vectors and θ be the angle between them then \(|\vec a-\vec b|\) isa. \(sin\cfrac{\theta}2\)b. 2 \(sin\cfrac{\theta}2\)c. 2 \(cos\cfrac{\theta}2\)d. \(cos\cfrac{\theta}2\)4. Let \(\vec a,\,\vec b\) and  \(\vec c\) be unit vectors such that \(\vec a.\vec b=\vec a.\vec c=0\) and angle between \(\vec b\) and  \(\vec c\) is \(\cfrac{\pi}6\) then \(\vec a\) =a. 2(\(\vec b\times\vec c\)). b. -2(\(\vec b\times\vec c\)) c. ±2(\(\vec b\times\vec c\)) d. 2(\(\vec b\pm\vec c\))5. The area of the parallelogram formed by \(\vec a\) and \(\vec b\) as diagonals isa. 70b. 35c. \(\cfrac{\sqrt{70}}2\)d. \(\sqrt{70}\) 
• Find the sum of the vector\(\vec{a} = \hat{i} - 2\hat{j} + \hat{k}\)
• Write the value of \((\hat{i}\times \hat{j}). \hat{k}+(\hat{j}\times \hat{k}).\hat{i}\)