AHL 3.13 (HL)—Scalar and vector products

The scalar product a·b=|a||b|cosθ measures alignment and is zero for perpendicular vectors. The vector product a×b is perpendicular to both and has magnitude |a||b|sinθ, the area of the parallelogram they span.

Use a·b for angles, projections and perpendicular tests; use a×b for normals, orientation and areas. Order matters for the cross product: b×a=−(a×b).

For a=(1,0,0) and b=(0,2,0), a·b=0 and a×b=(0,0,2), so the vectors are perpendicular and span area 2.

The cross product is not a scalar and the dot product is not a vector. Check the dimension and interpretation of the requested result.

The scalar component of $a$ in the direction of non-zero $b$ is $a\cdot b/|b|=|a|\cos heta$; the magnitude of the component perpendicular to $b$ in their plane is $|a\times b|/|b|=|a|\sin heta$. Thus $|a\times b|$ is parallelogram area and half of it is triangle area. For line directions, use the acute angle by replacing $\cos heta$ with $|a\cdot b|/(|a||b|)$.