The scalar product is one method of multiplying vectors which results in a scalar and has uses when working with vectors and lines. The vector product is a different method, which results in a vector and has uses when working with lines and planes.
IB DP Maths: AI HL复习笔记3.7.5 The Vector Product
The Vector ('Cross') Product
What is the vector (cross) product?
- The vector product (also known as the cross product) is a form in which two vectors can be combined together
- The vector product between two vectors v and w is denoted v × w
- The result of taking the vector product of two vectors is a vector
- The vector product is a vector in a plane that is perpendicular to the two vectors from which it was calculated
- This could be in either direction, depending on the angle between the two vectors
- The right-hand rule helps you see which direction the vector product goes in
- By pointing your index finger and your middle finger in the direction of the two vectors your thumb will automatically go in the direction of the vector product
How do I find the vector (cross) product?
- There are two methods for calculating the vector product
What properties of the vector product do I need to know?
Exam Tip
- The formulae for the vector product are given in the formula booklet, make sure you use them as this is an easy formula to get wrong
- The properties of the vector product are not given in the formula booklet, however they are important and it is likely that you will need to recall them in your exam so be sure to commit them to memory
Worked Example
ii)the formula , given that the angle between them is 1 radian.
Areas using Vector Product
How do I use the vector product to find the area of a parallelogram?
- The area of the parallelogram with two adjacent sides formed by the vectors vandw is equal to the magnitude of the vector product of two vectors vandw
How do I use the vector product to find the area of a triangle?
- The area of the triangle with two sides formed by the vectors vandw is equal to half of the magnitude of the vector product of two vectors vandw
Exam Tip
- The formula for the area of the parallelogram is given in the formula booklet but the formula for a triangle is not
- Remember that the area of a triangle is half the area of a parallelogram
Worked Example
Find the area of the triangle enclosed by the coordinates (1, 0, 5), (3, -1, 2) and (2, 0, -1).
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