System of Particles and Rotational Motion · Lesson 5 of 15
Vector Products and Their Geometry
“Compute cross products and connect their direction and magnitude with areas and volumes.”
• Apply the right-hand rule in the correct order. • Compute vector products from components. • Distinguish cross products from dot products. • Derive triangle area and scalar triple-product volume.
A product with a direction
The scalar product a · b = ab cosθ gives a scalar, such as work. The vector product a × b has magnitude ab sinθ and points perpendicular to the plane containing a and b. Curl the fingers of your right hand from a toward b through the smaller angle; your thumb gives the product direction. Equivalently, a right-handed screw turned from a toward b advances along a × b. For parallel or antiparallel vectors the product is zero, so no nonzero product direction is defined.
The cyclic unit-vector products are i × j = k, j × k = i and k × i = j. Reversing either order introduces a minus sign. The product of a vector with itself is zero. Expand component products using these rules, then collect the i, j and k terms. This explains the component formula rather than leaving it as a determinant to memorise.
Problem
Let a = (3, −4, 5) and b = (−2, 1, −3).
- 1.The dot product is 3(−2) + (−4)(1) + 5(−3) = −25.
- 2.The cross-product components are (12 − 5, −10 + 9, 3 − 8), giving (7, −1, −5).
- 3.Reversing order gives (−7, 1, 5). The dot product stays −25 because the dot product commutes.
Area and volume
A parallelogram with adjacent edges a and b has base a and perpendicular height b sinθ. Its area is ab sinθ = |a × b|. A triangle using those two edges occupies half that area. For a parallelepiped, take the b–c parallelogram as the base. Its area is |b × c| and the perpendicular height is the magnitude of a’s component along that base normal. Multiplying gives the magnitude of the scalar triple product.
Polar and axial behaviour
Under inversion of all spatial coordinates, ordinary position vectors reverse: a becomes −a and b becomes −b. Their cross product remains unchanged because (−a) × (−b) = a × b. A cross product is therefore an axial vector, also called a pseudovector. This statement concerns full spatial inversion; a mirror reflection in only one plane is not the same operation. Torque and angular momentum have this axial character.
Quiz
What is j × i?
Reverse the cyclic order i × j = k to obtain −k.
When is the magnitude of a × b greatest for fixed magnitudes?
sinθ reaches its maximum value 1 at 90°.
A triangle with edge vectors a and b has area what?
It occupies half the corresponding parallelogram.
Which is a scalar?
The outer operation is a dot product.
Under full inversion, a × b changes how?
Two sign reversals cancel in the bilinear cross product.
Key Takeaways
• Cross-product magnitude is ab sinθ; direction follows the ordered right-hand rule. • Reversing order reverses the result. • Area and volume follow from perpendicular projections. • Cross products have axial-vector behaviour under inversion.