JEE Mathematics Applications of Vectors in Mechanics MCQs

Question. P is a point in the plane of the \(\Delta ABC\) whose orthocenter is H and the circumcentre is O. Forces \(\vec{AP}, \vec{BP}, \vec{CP}\) and \(\vec{PH}\) act at P. The force that will keep the given forces in equilibrium is
(a) \(2\vec{OP}\)
(b) \(3\vec{OP}\)
(c) \(2\vec{PO}\)
(d) none of the options
Answer: (c) \(2\vec{PO}\)

Question. Three forces \(\vec{P}, \vec{Q}\) and \(\vec{R}\) each of 15 units, act along AB, BC and CA respectively. The position vectors of A, B and C are \(\vec{OA} = 2\vec{i} - \vec{j} + 3\vec{k}, \vec{OB} = 5\vec{i} + 3\vec{j} - 2\vec{k}\) and \(\vec{OC} = 2\vec{i} + 2\vec{j} + 3\vec{k}\). The resultant force vector is
(a) \(\left( \frac{9}{\sqrt{2}} - 7\sqrt{3} \right)\vec{i} - (9 - 6\sqrt{2} + \sqrt{3})\vec{j} + \left( 5\sqrt{3} - \frac{15}{\sqrt{2}} \right)\vec{k}\)
(b) \(\left( 12 + \frac{9}{\sqrt{2}} - 7\sqrt{3} \right)\vec{i} + (9 - 6\sqrt{2} + \sqrt{3})\vec{j} + \left( \frac{15}{\sqrt{2}} - 5\sqrt{3} \right)\vec{k}\)
(c) \(75\vec{i} + 60\vec{j} + 60\vec{k}\)
(d) none of the options
Answer: (a) \(\left( \frac{9}{\sqrt{2}} - 7\sqrt{3} \right)\vec{i} - (9 - 6\sqrt{2} + \sqrt{3})\vec{j} + \left( 5\sqrt{3} - \frac{15}{\sqrt{2}} \right)\vec{k}\)

Question. A ship is sailing towards north at a speed of 1.25 m/s. The current is taking it towards east at the rate of 1 m/s. A sailor is climbing a vertical pole on the ship at the rate of 0.5 m/s. The magnitude of the velocity of the sailor in space is
(a) 2.75 m/s
(b) \(\frac{3\sqrt{5}}{4}\) m/s
(c) \(\frac{3\sqrt{5}}{2}\) m/s
(d) none of the options
Answer: (b) \(\frac{3\sqrt{5}}{4}\) m/s

Question. A force \(10\vec{i} - 5\vec{j} + 7\vec{k}\) displaces a particle from the point A to the point B. The position vectors of A and B are \(3\vec{i} - \vec{j} + 2\vec{k}\) and \(\vec{i} + 3\vec{j} + 2\vec{k}\) respectively. Then the work done is
(a) 40
(b) 20
(c) 60
(d) none of the options
Answer: (a) 40

Question. Constant forces \(\vec{P} = \vec{i} - 2\vec{j} + 3\vec{k}, \vec{Q} = -\vec{i} + 3\vec{j} - \vec{k}\) and \(\vec{R} = 2\vec{i} - 4\vec{j} + 3\vec{k}\) act on a particle. The work done from the point \(4\vec{i} - 3\vec{j} - 2\vec{k}\) to the point B with position vector \(6\vec{i} + \vec{j} - 3\vec{k}\) is
(a) 15
(b) 13
(c) \(\sqrt{13}\)
(d) none of the options
Answer: (b) 13

Question. The vertices of a triangle ABC are A(–1, 0, 2), B(1, 2, 0) and C(2, 3, 4). The moment of a force of magnitude 10 acting at A along AB about C is
(a) \(\frac{50\sqrt{6}}{3}\)
(b) \(20\sqrt{6}\)
(c) \(\frac{50}{\sqrt{3}}\)
(d) none of the options
Answer: (a) \(\frac{50\sqrt{6}}{3}\)

Question. The vector moment about the point \(\vec{i} + 2\vec{j} + 3\vec{k}\) of the resultant of the forces \(\vec{i} - 2\vec{j} + 5\vec{k}\) and \(3\vec{j} - 4\vec{k}\) acting at the point \(2\vec{i} + 3\vec{j} - \vec{k}\) is
(a) \(5\vec{i} + \vec{j} - 4\vec{k}\)
(b) \(5\vec{i} - \vec{j} - 4\vec{k}\)
(c) \(3\vec{i} + \vec{j} - 4\vec{k}\)
(d) none of the options
Answer: (b) \(5\vec{i} - \vec{j} - 4\vec{k}\)

Question. A rigid body is rotating at 5 radians per second about an axis AB, where A and B are points whose position vectors are \(2\vec{i} + \vec{j} + \vec{k}\) and \(8\vec{i} - 2\vec{j} + 3\vec{k}\) respectively. The velocity of the particle of the body at the point whose position vector is \(5\vec{i} + 2\vec{j} - \vec{k}\) is
(a) \(\frac{15}{7}(2\vec{i} + 6\vec{j} + 3\vec{k})\)
(b) \(\frac{5}{7}(4\vec{i} - 6\vec{j} - 3\vec{k})\)
(c) \(\frac{15}{7}(4\vec{i} - 6\vec{j} - 3\vec{k})\)
(d) none of the options
Answer: (a) \(\frac{15}{7}(2\vec{i} + 6\vec{j} + 3\vec{k})\)

Question. A particle is in equilibrium when the forces \(\vec{F}_1 = -10\vec{k}, \vec{F}_2 = \frac{u}{13}(4\vec{i} - 12\vec{j} + 3\vec{k}), \vec{F}_3 = \frac{v}{13}(-4\vec{i} - 12\vec{j} + 3\vec{k})\) and \(\vec{F} = w(\cos \theta \vec{i} + \sin \theta \vec{j})\) act on it. Then
(a) \(u = 65(1 - 3\cot \theta)\)
(b) \(v = \frac{65}{3} + 65\cot \theta\)
(c) \(w = 40\text{cosec } \theta\)
(d) none of the options
Answer: (b) \(v = \frac{65}{3} + 65\cot \theta\), (c) \(w = 40\text{cosec } \theta\)

Question. The resolved parts of the force vector \(5\vec{i} + 4\vec{j} + 2\vec{k}\) along and perpendicular to the vector \(3\vec{i} + 4\vec{j} - 5\vec{k}\) are \(\vec{u}\) and \(\vec{v}\) respectively. Then
(a) \(\vec{u} = \frac{21}{50}(3\vec{i} + 4\vec{j} - 5\vec{k})\)
(b) \(\vec{u} = \frac{21}{5\sqrt{2}}(3\vec{i} + 4\vec{j} - 5\vec{k})\)
(c) \(\vec{v} = \frac{1}{50}(187\vec{i} + 116\vec{j} + 5\vec{k})\)
(d) \(\frac{1}{5\sqrt{2}}(\vec{i} - 2\vec{j} - \vec{k})\)
Answer: (a) \(\vec{u} = \frac{21}{50}(3\vec{i} + 4\vec{j} - 5\vec{k})\), (c) \(\vec{v} = \frac{1}{50}(187\vec{i} + 116\vec{j} + 5\vec{k})\)