Solve problems using i, j and k unit vectors

Solve problems using i, j and k unit vectors

Understanding i, j, and k Unit Vectors

  • i, j, and k are standard unit vectors in 3D space representing the x, y, and z dimensions respectively.
  • A unit vector has a magnitude (or length) of one and describes a direction.
  • Unit vectors can be used to represent any vector in space through a directed line segment.
  • The direction of i is along the x-axis, j along the y-axis, and k along the z-axis, while their magnitudes are all equal to 1.

Using Unit Vectors

  • The position vector of a point (x, y, z) in 3D space can be represented as xi + yj + zk.
  • To represent a vector in 3D space, the relevant coefficients of the unit vectors i, j, and k are found, effectively breaking down the vector into its components along the axes.
  • A vector operation (like addition, subtraction, etc.) can be carried out by performing the operation on the individual components of the vectors.
  • Scalar multiplication can be carried out by multiplying each component of the vector by the scalar.

Solving Problems

  • To add or subtract vectors, add or subtract the corresponding components. If A = ai + bj + ck and B = di + ej + fk, then A + B = (a+d)i + (b+e)j + (c+f)k.
  • Dot product of two vectors A and B is calculated by multiplying the corresponding components and adding them up: A.B = ai.bj + aj.bj + ak.bk.
  • The cross product of two vectors results in a third vector that is perpendicular to the plane containing the first two vectors. If A = ai + bj + ck and B = di + ej + fk, then A x B = (bf - ce)i - (af - cd)j + (ae - bd)k.
  • To find the magnitude of a vector represented with unit vectors, square each of the component, add them up, and take the square root.
  • To find the angle between two vectors, use the dot product formula: A.B =   A       B   cos(theta), where theta is the angle between A and B.
  • When tackling physics problems, use i, j, and k to represent forces, velocities, and accelerations in their respective components along the axes. Then apply the appropriate vector or scalar operation.