Vectors Formulae (Shortest Distance, Angle Between)

Point Line

• Take the vector from a point on the line to your point.

• Cross it with the line’s direction vector.

• Divide the length of that cross product by the length of the line’s direction vector.

• ✅ Gives the shortest (perpendicular) distance.

Skew Lines

• Take the vector connecting any point on line 1 to any point on line 2.

• Dot it with the cross product of the two lines’ direction vectors.

• Take the absolute value and divide by the length of the cross product of the directions.

• ✅ Gives the shortest distance between the lines.

Angle Line and plane

• Take the line’s direction vector.

• Dot it with the plane’s normal vector.

• Divide by the product of their lengths.

• Take the arcsine of that value.

• ✅ This gives the angle between the line and the plane.

Angle Plane and plane

• Take the normals of the two planes.

• Dot them together and divide by the product of their lengths.

• Take the arccosine of that value.

• ✅ This gives the angle between the planes.

Perpendicular distance between a line and a plane in parallel

• Pick any point $P(x_0, y_0, z_0)$ that lies on the line (this is usually given as the position vector in the line's equation).

• Ensure the plane equation is in Cartesian form: $ax + by + cz + d = 0$.

• Substitute the point's coordinates into the plane's expression: $ax_0 + by_0 + cz_0 + d$.

• Take the absolute value of that result and divide it by the magnitude (length) of the normal vector $\sqrt{a^2 + b^2 + c^2}$.

• ✅ This gives the perpendicular distance $D$.

Distance between parallel planes

1. Make sure both planes are written in the form

   ax+by+cz+d=0ax+by+cz+d=0

2. Check that the x,y,zx,y,z coefficients are the same.
   ✅ This confirms the planes are parallel.

3. Subtract the constant terms.

   ∣d2−d1∣∣d2​−d1​∣

4. Find the magnitude of the normal vector.

   a2+b2+c2a2+b2+c2​

5. Divide the difference in constants by the magnitude of the normal vector.

6. ✅ This gives the perpendicular distance between the planes.