Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — the angle between two vectors
Find the angle between a=2i+3j−k and b=i−2j+4k.
- 1a⋅b=(2)(1)+(3)(−2)+(−1)(4)Multiply matching components and add.
- 2=2−6−4=−8Negative, so the angle will be obtuse.
- 3∣a∣=4+9+1=14
- 4∣b∣=1+4+16=21
- 5cosθ=1421−8=294−8=−0.4666Between −1 and 1, as it must be.
- 6θ=117.8° (1 d.p.)Obtuse, exactly as the negative product predicted ✓
Example 2 — the vector equation of a line through two points
Find the vector equation of the line through A(1,2,3) and B(3,1,7).
- 1A line needs a point and a direction.
- 2Take the point A: a=i+2j+3kEither point works.
- 3Direction AB=b−a=2i−j+4kDestination minus start.
- 4r=(i+2j+3k)+t(2i−j+4k)
- 5Check: t=1 gives 3i+j+7k, which is B ✓Starting from B instead would give a different-looking but equally valid answer.
Example 3 — testing for perpendicularity
Show that 2i+3j−k and i−j−k are perpendicular.
- 1Two non-zero vectors are perpendicular exactly when their scalar product is zero.Because cos90°=0.
- 2(2)(1)+(3)(−1)+(−1)(−1)Careful with the double negative in the last term.
- 3=2−3+1
- 4=0So the vectors are perpendicular. No angle calculation needed.