Physics · Motion In A Plane · NEET
Each of them has magnitude exactly 1. That is the whole point of a 'unit' vector: |i| = |j| = |k| = 1. They carry no size, only direction. So in the vector 5i, the length is 5 (not 6), because i itself is just length 1 pointing along the x-axis and the 5 scales it.
In 3i, the '3' is a plain number (a scalar) that tells you the size, and 'i' is the unit vector that tells you the direction (along +x). So 3i means 'go 3 units in the x direction'. If a vector is A = Ax i + Ay j + Az k, then Ax, Ay, Az are just numbers (the components), and i, j, k are the direction markers.
Divide the vector by its own magnitude. The unit vector is n = A / |A|. Example: for A = 3i + 4j, |A| = sqrt(3^2 + 4^2) = 5, so n = (3i + 4j)/5 = 0.6 i + 0.8 j. Check: |n| = sqrt(0.6^2 + 0.8^2) = 1. This 'normalising' trick is how you build a unit vector in any direction you need.
They point along the x, y, z axes but they are vectors, not the axes themselves. i points in the +x direction, j in the +y direction, k in the +z direction, in a right-handed system. The axes are lines; i, j, k are unit arrows sitting on those lines to give direction.
The hat (as in i-hat) is the standard mark for a unit vector — a vector whose magnitude is 1. When you see any letter with a hat, read it as 'the direction only, length 1'. So r-hat means the unit vector pointing along r.
The position of a particle is given by r(t) = 4t i + 2t^2 j + 5 k (metres, t in seconds). Find the magnitude and direction (with respect to the x-axis) of the velocity v(t) at t = 1 s.
A particle moves from point (-2 i + 5 j) to (4 j + 3 k) when a force (4 i + 3 j) N is applied. How much work is done by the force?
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Almost every Motion in a Plane, Work-Energy, and Magnetism numerical gives forces, velocities and positions in i, j, k form. If you are comfortable reading Ax i + Ay j + Az k, you can instantly pull out components, take dot and cross products, and find magnitudes — which is exactly what NEET tests.
No. They are pure direction markers with magnitude 1 and no units. The units come from the number in front. In v = 4 i + 4 j m/s, the 'm/s' belongs to the components 4 and 4, not to i and j.
i · i = 1 (a unit vector dotted with itself gives 1, since cos 0° = 1). i · j = 0 (they are perpendicular, cos 90° = 0). The same holds for j and k. These simple results are why the dot product of two vectors in component form just multiplies matching components and adds them.
i × j = k, j × k = i, k × i = j (cyclic order in a right-handed system). Reverse the order and you get a minus sign, e.g. j × i = -k. Also i × i = 0. These rules let you compute the cross product of any two vectors written in i, j, k form.
Only i and j are needed for a plane. A vector A at angle θ to the x-axis with magnitude A is A = A cos θ i + A sin θ j. The next step, rectangular components, uses exactly this to break any planar vector into its x and y parts.
