MATH1131 Chap.2 Dot Product, Cross Product and Projections
Dot Product, Cross Product and Projections
The dot product turns two vectors into a number and carries most of the geometry in the course: lengths come from a vector dotted with itself, angles from dividing the dot product by the two lengths, and right angles from a zero dot product. Projection of one vector onto another uses the same quantity divided by the squared length of the second vector, and it underlies the shortest distance from a point to a line or a plane.
The cross product is special to three dimensions. It produces a vector perpendicular to both factors whose length equals the area of the parallelogram they span, so it supplies normal vectors for planes and areas of triangles. Combining the two products gives the scalar triple product, whose absolute value is the volume of a parallelepiped and whose vanishing signals that three vectors lie in a plane.
The chapter closes with planes in three dimensions written using a normal vector, and with the point-to-plane distance formula that follows from projecting onto that normal.
What this chapter covers
- 01
Length and distance in R^n
- 02
Dot product and the angle between vectors
- 03
Orthogonality and orthonormal sets
- 04
Projection of one vector on another
- 05
Cross product and the right-hand rule
- 06
Areas of parallelograms and triangles
- 07
Scalar triple product and volume
- 08
Point-normal form of a plane and point-to-plane distance
Worked example · free
Distance from a point to a line using projection
- 1Take A(1, 0, 0) on the line and form AB = (3, 2, 1); the direction is d = (1, 1, 0) with |d|^2 = 2.
- 1Projection of AB on d: (AB . d / |d|^2) d = (5/2)(1, 1, 0) = (5/2, 5/2, 0).
- 1Perpendicular part: AB minus the projection = (1/2, -1/2, 1), which is orthogonal to d since 1/2 - 1/2 + 0 = 0.
- 1Distance = |(1/2, -1/2, 1)| = sqrt(1/4 + 1/4 + 1) = sqrt(3/2), about 1.225.
Key terms
- Dot product
- The scalar formed by multiplying corresponding components of two vectors and adding the results.
- Orthogonal vectors
- Vectors whose dot product is zero, meaning they meet at a right angle when both are non-zero.
- Projection
- The component of one vector that lies along another, given as a multiple of the second vector.
- Cross product
- A vector in three dimensions perpendicular to both factors, with length equal to the area they span.
- Scalar triple product
- The number formed by dotting one vector with the cross product of two others, equal in size to a parallelepiped volume.
- Normal vector
- A non-zero vector perpendicular to every direction lying in a given plane.
Dot Product, Cross Product and Projections FAQ
When should I use the dot product rather than the cross product?
Use the dot product for angles, orthogonality tests, lengths and projections. Use the cross product when you need a vector perpendicular to two given vectors, a normal to a plane, or the area spanned by two vectors in three dimensions.
Why is |b| squared in the projection formula?
One factor of |b| converts the dot product into a length along the direction of b, and the other normalises b so the result is a vector of that length pointing along b.
Does the order of factors in a cross product matter?
Yes. Reversing the order reverses the resulting vector, so b cross a is the negative of a cross b. Lengths and areas are unaffected, but a normal vector or orientation would point the other way.
What does a zero scalar triple product tell me?
It means the three vectors are coplanar: the parallelepiped they define is flat and has no volume. Equivalently, the 3 by 3 determinant with those vectors as rows is zero.
Is the point-to-plane distance ever negative?
No. The signed quantity before taking the absolute value only tells you which side of the plane the point lies on relative to the chosen normal; the distance itself is its absolute value.
How do I find the angle between two planes?
Find a normal vector for each plane from its Cartesian coefficients, then use the dot product formula for the angle between those normals. The acute angle between the planes equals the acute angle between the normals.
Exam move
Build a one-page sheet with the dot product, angle formula, projection, cross product components, area and volume formulas, each with one small numerical example you have verified yourself. Then practise the multi-step problems that combine them, such as distances from points to lines and planes, because those are where the exam asks for explanation as well as calculation.
After each cross product, dot it with both factors to check for zeros. Time yourself on a few problems from the Algebra Notes so the arithmetic becomes quick enough to leave room for written reasoning. Keep exact surds such as the square root of three until the final line, and only then give a decimal if one is asked for; rounding early makes later checks against orthogonality fail by small amounts.
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