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: AA HL复习笔记3.10.4 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?
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?
Exam Tip
- The formula for the area of the parallelogram is given in the formula booklet but the formula for the area of 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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