Basics of Vectors

Problems Based on Fundamentals of Vectors:

• The minimum number of coplanar vectors with different magnitudes needed to achieve a zero resultant is 3.
• A fly travels from one corner of a hall to the diagonally opposite corner. The hall dimensions are 10m x 12m x 14m, and the magnitude of the displacement is 21m.
• If 0.4î + 0.8ĵ + ck represents a unit vector, then c is √0.2.
• When 100 coplanar forces, each of 10N, act on a body with each force at an angle of π/50 to the preceding one, the resultant force is 0.
• Given a vector with endpoints (4, -4, 0) and (-2, -2, 0), the magnitude is 2√10.
• A vector î + ĵ + √2k makes angles of 45°, 45°, and 60° with the X, Y, and Z axes, respectively.
• The expression (1/√2)î + (1/√2)ĵ is a unit vector.
• Given vector A = 2î + 3ĵ, the angle between A and the y-axis is tan⁻¹(2/3).
• The unit vector along î + ĵ is (î + ĵ)/√2.
• The length in the XY plane of a vector represented by 3î + ĵ + 2k is √10.
• If equal forces of 10 N each are applied at one point and all lie in one plane with equal angles between them, the resultant force will be zero.
• The angle that the vector A = î + ĵ makes with the x-axis is 45°.
• The value of a unit vector in the direction of vector A = 5î - 12ĵ is (5î - 12ĵ)/13.
• Any vector in an arbitrary direction can be replaced by: Parallel vectors which have the original vector as their resultant, Mutually perpendicular vectors which have the original vector as their resultant and Arbitrary vectors which have the original vector as their resultant.
• Angular momentum is an axial vector.
• If a vector P makes angles α, β, and γ with the X, Y, and Z axes respectively, then sin²α + sin²β + sin²γ = 2.

Problems Based on Addition of Vectors:

• For two forces, each of magnitude F, to have a resultant of the same magnitude F, the angle between them is 120°.

• For the resultant of two vectors to be maximum, the angle between them must be 0°.

• If a particle is simultaneously acted upon by two forces equal to 4N and 3N, the net force on the particle is between 1N and 7N.

• Given two vectors A and B lie in a plane, and another vector C lies outside this plane, then the resultant of these three vectors A+B+C cannot be zero.

• If the resultant of two forces has a magnitude smaller than the magnitude of the larger force, the two forces must be different in magnitude and have an obtuse angle between them.

• For forces F₁ and F₂ acting on a point mass in two mutually perpendicular directions, the resultant force on the point mass will be √(F₁² + F₂²).

• If |A - B| = |A| = |B|, the angle between A and B is 120°.

• For two forces (x+y) and (x-y) to have a resultant of √(x² + y²), the angle must be cos⁻¹((x² + y²)/(2(x² - y²))).

• If the angle between two nonzero vectors A and B is 120° and the resultant is C, then C must be lessthan |A - B|.

• In a regular hexagon ABCDEF, AB + AC + AD + AE + AF = 6 AO.

• If the magnitudes of vectors A, B, and C are 12, 5, and 13 units respectively, and A + B = C, then the angle between A and B is π/2.

• The magnitude of the vector resulting from the addition of two vectors, 6î + 7ĵ and 3î + 4ĵ, is √136.

• A particle has a displacement of 12m towards east, 5m towards north, and 6m vertically upward. The sum of these displacements is None of these.

• The three vectors Ā = 3î - 2ĵ + k, B = î - 3ĵ + 5k, and Ĉ = 2î + ĵ - 4k form a right-angled triangle.

• If Ĉ = A + B, it is possible to have |Ĉ| < |A| and |Ĉ| < |B|.

• The value of the sum of two vectors A and B, with θ as the angle between them, is √(A² + B² + 2AB cos θ).

• Given forces acting simultaneously on a particle at rest at the origin of the co-ordinate system: F₁ = -4î - 5ĵ + 5k, F₂ = 5î + 8ĵ + 6k, F₃ = -3î + 4ĵ - 7k, and F₄ = 2î - 3ĵ - 2k, then the particle will move in the y-z plane.

• The following sets of three forces act on a body; resultant cannot be zero when 10, 20 and 40.

• When three forces of 50N, 30N, and 15N act on a body, then the body is moving with acceleration.

• The sum of two forces acting at a point is 16N and the resultant force is 8N, with its direction perpendicular to the minimum force, then the forces are 6N and 10N.

• If vectors P, Q, and R have magnitudes 5, 12, and 13 units, and P + Q = R, then the angle between Q and R is cos⁻¹(12/13).

• The resultant of two vectors A and B is perpendicular to vector A, and its magnitude is equal to half the magnitude of vector B. The angle between A and B is 150°.

• To achieve a unit vector along the x-axis by adding a vector to the two vectors î - 2ĵ + 2k and 2î + ĵ - k, the vector to be added must be -2î + ĵ - k.

• The angle between P and the resultant of (P+Q) and (P-Q) is zero.

• If the resultant of P and Q is perpendicular to P, then the angle between P and Q is cos⁻¹(-P/Q).

• If the maximum and minimum magnitudes of the resultant of two vectors P and Q are in the ratio 3:1, then P = 2Q.

• If the resultant of A + B is R₁, and on reversing vector B, the resultant becomes R₂, then the value of R₁² + R₂² is 2(A² + B²).

• If the resultant of two vectors P and Q is R, and if Q is doubled, the new resultant is perpendicular to P, then R equals Q.

• Two forces, F₁ and F₂, are acting on a body, where one force is double the other force, and the resultant is equal to the greater force. Then the angle between the two forces is cos⁻¹(-1/4).

• Given A + B = C, and that C is perpendicular to A, with |A| = |C|, then the angle between A and B is 3π/4 radian.

Problems Based on Subtraction of Vectors:

• Given a body of mass M moving with uniform speed on a circular path with radius R. The change in acceleration, in going from P₁ to P₂, is v²/R x √2.
• If a body is at rest under the action of three forces, with two forces being F₁ = 4î and F₂ = 6ĵ, then the third force is -4î - 6ĵ.
• A plane revolving around the Earth with a speed of 100 km/hr at a constant height experiences a velocity change of 200 km/hr as it travels half a circle.
• To achieve a displacement of 7.0 m pointing in the x-direction by adding a displacement to 25î - 6ĵm, then the required displacement must be -18î + 6ĵ.
• If a body moves due East with a velocity of 20 km/hr and then due North with a velocity of 15 km/hr, the resultant velocity is 25 km/hr.
• A particle moving on a circular path of radius r with uniform velocity v experiences a change in velocity of 2v sin 20° when it moves from P to Q with an angle ∠POQ = 40°.
• The length of the seconds hand in a watch is 1 cm. The change in velocity of its tip in 15 seconds is (π√2)/30 cm/sec.
• A particle moving towards east with a velocity of 5 m/s changes direction towards north with the same velocity after 10 seconds. The average acceleration of the particle is (1/√2) m/s² S-W.

Problems Based on Scalar Product of Vectors:

• Given two vectors F₁ = 2î + 5k and F₂ = 3ĵ + 4k, the scalar product of these vectors is 20.

• Given a vector F = 4î - 3ĵ, another vector perpendicular to F is 7k.

• Two vectors A and B are at right angles to each other when A.B = 0.

• If |V₁ + V₂| = |V₁ - V₂| and V₂ is finite, then V₁ and V₂ are mutually perpendicular.

• A force F = (5î + 3ĵ) Newton is applied over a particle displacing it from its origin to the point r = (2î - ĵ) metres. The work done on the particle is +7 joules.

• The angle between two vectors -2î + 3ĵ + k and î + 2ĵ - 4k is 90°.

• The angle between the vector (î + ĵ) and (ĵ + k) is 60°.

• A particle moves with a velocity of 6î - 4ĵ + 3k m/s under the influence of a constant force, F = 20î + 15ĵ - 5k kN. The instantaneous power applied to the particle is 45 J/s.

• If P.Q = PQ, then the angle between P and Q is 0°.

• If two constant forces F₁ = 2î - 3ĵ + 3k (N) and F₂ = î + ĵ - 2k (N) act on a body and displace it from the position r₁ = î + 2ĵ - 2k (m) to the position r₂ = 7î + 10ĵ + 5k (m). What is the work done is 41J.

• A force F = 5î + 6ĵ + 4k acting on a body produces a displacement s = 6î - 5k. The work done by the force is 10 units.

• The angle between the two vectors A = 5î + 5ĵ and B = 5î - 5ĵ will be 90°.

• The vectors P = aî + aĵ + 3k and Q = aî - 2ĵ - k are perpendicular to each other. The positive value of a is 3.

• A body constrained to move in the Y direction is subjected to a force given by F = (-2î + 15ĵ + 6k) N. What is the work done by this force in moving the body a distance of 10m along the Y-axis? 150 J.

• A particle moves in the x-y plane under the action of a force F such that its linear momentum (P) at any time t is Px = 2 cost, Py = 2 sint. The angle θ between F and P at a given time t will be 90°.

Problem Based on Cross Product of Vectors:

• The area of the parallelogram represented by the vectors A=2î+3ĵ and B= î+4ĵ is 5 units.

• For any two vectors A and B, if A⋅B = |A × B|, the magnitude of (A + B) is equal to √(A² + B² + √2 × AB).

• A vector F is along the positive x-axis. Its vector product with another vector F₂ is zero, then F₂ could be -4î.

• If for two vectors A and B, A × B = 0, the vectors are parallel to each other.

• The angle between vectors (A × B) and (B × A) is π.

• The angle between (P + Q) and (P × Q) is π/2.

• For vectors with magnitudes 2 and 3, with a resultant magnitude of 1, the magnitude of their cross product is 0.

• The unit vector perpendicular to A and B is (A × B) / (AB sin θ).

• Let A = iAcosθ + jAsinθ be any vector. Another vector B, which is normal to A, is iBsinθ - jBcosθ.

• The angle between the two vectors given by 6î + 6ĵ - 3k and 7î + 4ĵ + 4k is sin⁻¹(√5/3).

• If vector A points vertically upwards and vector B points towards the north, then A × B is along the west.

• The angle between the vectors (î + ĵ) and (ĵ - k) is 60°.

• Two vectors P = 2î + bj + 2k and Q = î + ĵ + k will be parallel if b = 1.

• For position vectors of points A, B, C and D; A = 3î + 4ĵ + 5k, B = 4î + 5ĵ + 6k, C = 7î + 9ĵ + 3k, and D = 4î + 6ĵ, the displacement vectors AB and CD are antiparallel.

• For A = 3î + 4ĵ and B = 6î + 8ĵ, where A and B are the magnitudes, is not true if A = 5.

• If force F = 4î + 5ĵ and displacement S = 3î + 6k, then the work done is 4 x 3.

• If |A × B| = |A.B|, then the angle between A and B will be 90°.

• The correct option is j × k = î.

• The linear velocity of a rotating body is given by v = w × r, and w = î - 2ĵ + 2k and r = 4ĵ - 3k, then |v| is √29 units.

• For the three vectors a, b, and c that satisfy the relation a.b = 0 and a.c = 0, the vector a is parallel to b × c.

• The diagonals of the area of a parallelogram are 2î and 2ĵ what is the area of the parallelogram 2 units.

• The unit vector perpendicular to the vectors 2î + 2ĵ - k and 6î - 3ĵ + 2k is (î - 10ĵ - 18k) / (5√17).

• The area of the parallelogram with sides represented by the vectors ĵ + 3k and î + 2ĵ - k is √59 sq. units.

• The position of a particle is given by r = (î + 2ĵ - k), with momentum P = (3î + 4ĵ - 2k). The angular momentum is perpendicular to the x-axis.

• For two vectors, A and B, having equal magnitudes, the vector (A + B) is perpendicular to (A × B).

• The torque of force F = -3î + ĵ + 5k acting at the point r = 7î + 3ĵ + k is 14î - 38ĵ + 16k.

• The value of (A + B) × (A - B) is 2(B × A).

• A particle of mass m = 5 is moving with uniform speed v = 3√2 in the XOY plane along the line Y = X + 4. The magnitude of the angular momentum of the particle about the origin is 60 units.