IIT-JEE Physics Daily Practice Problems

Vector Operations and Properties

• The vector from point A (1, 1, -1) to point B (2, -3, 4) is calculated by subtracting the coordinates, resulting in: î + 4ĵ + 5k̂.
• The resultant vector of two vectors a and b, inclined at an angle θ, changes when their directions are interchanged; however, the magnitude remains the same.

Resultant Forces and Magnitudes

• Vector sum of forces can vary based on direction:
  • Example: Forces of 10 N and 6 N can yield resultant values of 2 N, 8 N, 18 N, or 20 N depending on the angle between them.
• Given magnitudes A (12), B (5), and C (13), the angle between A and B can be derived from the law of cosines.

Vector Relations and Properties

• The sum and difference of two perpendicular vectors of equal lengths yields resultant vectors that are also perpendicular and of equal length.
• For three non-zero vectors A, B, and C, if A is perpendicular to both B and C (A·B = 0 and A·C = 0), A may be parallel to the cross product of B and C.

Scalar and Vector Product

• The magnitude of the scalar product of two vectors being 8 and the vector product being (8\sqrt{3}) leads to the calculation of the angle between them—possible angles include 30º, 60º, 120º, or 150º.

Displacement and Return Points

• A set of displacements that returns the car to its starting point must satisfy the triangle inequality.
• Example displacements to check include various combinations varying from 5 km to 200 km.

Vector Direction and Cross Product

• A vector A pointing downward and vector B pointing east results in a vector product A × B that is directed west.

Equality of Vectors

• In a closed triangle formation (a + b + c = 0), when two vectors are of equal magnitude and the third vector's magnitude is double, the angles between them can vary; possible configurations include angles of 90º, 135º, and 60º.

Displacement in Three-Dimensional Space

• The displacement magnitude from one corner of a hall with dimensions 10 m × 12 m × 14 m to the opposite corner can be calculated using the Pythagorean theorem, resulting in a magnitude close to 17 m.