The CROSS X PRODUCT
(Geometric Vectors)
A.
The CROSS PRODUCT DEFINED
The cross product, 𝑢 x 𝑣, gives a vector which is  to both 𝑢 and 𝑣.
There are two vectors that have this property – they are equal in magnitude but
opposite in direction.
𝑢
𝑢x𝑣
𝜃
𝑣
𝜃
𝑣
𝑢x𝑣
𝑢
“upwards”
“downwards”
The cross product (or vector product) of two vectors is defined as:
Note: 𝑢 x 𝑣 = − 𝑣 x 𝑢
𝒖 𝐱 𝒗 = 𝒖 𝒗 𝒔𝒊𝒏𝜽
where 0o    180o
Ex.
Determine 𝑢 x 𝑣 and state the direction of 𝑢 𝑥 𝑣 if:
𝑢 = 3, 𝑣 = 5, 𝜃 = 40𝑜
𝑢
40o
𝑣
B.
APPLICATIONS of the CROSS PRODUCT
Consider the parallelogram defined by vectors, 𝑎 and 𝑏:
AREA of a PARALLELOGRAM:
A = 𝑎 𝑏 𝑠𝑖𝑛𝜃
= 𝑎x𝑏
Ex.
Calculate the area of the parallelogram formed by the vectors, 𝑢 and 𝑣,
where 𝑢 = 3, 𝑣 = 8, and 𝜃 = 63𝑜 .
𝑢
𝑣