A Set of 3D Vector Problems for Grade 10 IB

Basic 3D Vector Operations

1. Vector Addition and Subtraction:
   • Given vectors a = (2, -1, 3) and b = (-3, 4, 0), find a + b and a - b.
   • Find the magnitude and direction of the resultant vector a + b.
2. Scalar Multiplication:
   • If c = (1, 2, -4) and k = 2, find kc.
   • Determine the unit vector in the direction of c.
3. Dot Product:
   • Find the dot product of a = (3, -2, 1) and b = (1, 4, -2).
   • Determine the angle between vectors a and b.

Vector Equations of Lines and Planes

1. Vector Equation of a Line:
   • Find the vector equation of a line passing through the points A(1, 2, 3) and B(4, -1, 2).
   • Determine the parametric equations of the line.
2. Vector Equation of a Plane:
   • Find the vector equation of a plane passing through the points A(1, 2, 3), B(4, -1, 2), and C(2, 0, 1).
   • Find the normal vector to the plane.
3. Intersection of Lines and Planes:
   • Find the point of intersection of the line r = (1, 2, 3) + t(2, -1, 4) with the plane 2x - y + 3z = 5.

https://effectuall.github.io/Simulations/Mechanics_Vector_Display.html

https://effectuall.github.io/Simulations/Mechanics_Vector_Algebra

Vector Applications

1. Forces and Resultants:
   • A force of 20 N acts along the vector a = (1, 2, 3) and a force of 30 N acts along the vector b = (2, -1, 4). Find the resultant force and its direction.
2. Velocity and Displacement:
   • A particle moves with a velocity of v = (2t, 3t^2, 4t^3) m/s. Find the displacement of the particle between t = 0 and t = 2 seconds.

Challenging Problems

1. Vector Projections:
   • Find the projection of vector a = (3, -2, 1) onto vector b = (1, 4, -2).
2. Cross Product:
   • Find the cross product of a = (1, 2, 3) and b = (2, -1, 4).
   • Use the cross product to find the area of a parallelogram with sides a and b.
3. Vector Triple Product:
   • Evaluate the vector triple product a × (b × c) where a = (1, 2, 3), b = (2, -1, 4), and c = (3, 0, -2).

These problems extend the concepts of vector algebra to three dimensions, providing a more comprehensive understanding of vectors and their applications. They are suitable for IB Grade 10 students and can be adjusted in difficulty based on the students' understanding.