# VECTORS IN THREE DIMENSIONS SS2 FURTHER MATHS SECOND TERM

WEEK FIVE

FURTHER MATHEMATICS, SS2, SECOND TERM ,

SS2 FURTHER MATHS SECOND TERM

VECTORS IN THREE DIMENSIONS

CONTENTS

(a) Vectors or cross product in three dimensions and Properties of vector product.

(b) Utility of cross product.

SUB TOPIC: VECTOR OR CROSS PRODUCT AND PROPERTIES

If a, b and c are three vectors, then ax(bxc) and (axb)xc are referred to as the vector.

Note: ax(bxc) ≠ (axb)xc

Vector product is known as cross product or outer product. It produces vector.

Properties of Vector Product

thus the vector product of two vectors is just not commutative.

1. If a x b = 0 and a and b are non zero vectors, then a and b are parallel.
2. If

Examples:

1. Discover the vector product of a and b the place: a =4i – 3j + ok and b = -I + 2j + 3k
2. Therefore discover |a x b| of the query above.

1. |a x b| = |

CLASS ACTIVITY:

1. What can we imply by vector or cross product
2. _____ vectors are vectors on the identical airplane
3. Ax(BxC) is _____ mixture of B and C.
4. If a =3i – 5j + 2k and b = 4i – 5j +3k, discover a x b and |axb|
5. Checklist three properties of cross vector.

SUB TOPIC: APPLICATIONS OF CROSS PRODUCT

Vector product as quantity

Vector merchandise can be utilized to find out the quantity of a parallelepiped. If three vectors a, b, c characterize the size, width and peak of a parallelepiped, the quantity of the parallelepiped is given by the scalar triple product of a,b,c.

V = (axb).c

Examples

1. Discover the quantity of a parallelepiped whose vectors that are the edges given a = 2i + j + ok, b = i -3j +2k, c=3i+2j-k.

Quantity of the parallelepiped is given as

V = (axb).c

However (axb).c = a.(bxc)

a.(bxc) = =2 = 2(3-4) – (-1-6)+ (2+9)

= – 2 + 7 + 11 = 16 items

1. Discover the quantity of parallelepiped whose vectors which sides are given as a = 3i + j + 2k, b = i -2j + 3k, c = 4i + j – 2k

However (axb).c = a.(bxc)

a.(bxc) = = 3

= 3(4-3) – (-2-12) + (1+9) = 3+14+18= 35 items

CLASS ACTIVITY

1. If Ã = 3i-j+2k, = 2i +j – ok and = i – 2j + 2k. Discover (a) (Ãx) x (b) Ãx x )
2. Find the volume of a parallelepiped whose vectors that are the edges given x = 2i +j+ok, y=i-3j+2k, z = 3i + 2j –ok

PRACTICE EXERCISE

1. a,b,c are three vectors vectors, a = -6i+2j+ok, b= 3i-2j+4k, c=5i+7j+3k. Discover ax(bxc).
2. If a = 2i+2j+3k, b =-1+2j+ok and c=3i+j. Discover (axb)xc.
3. Outline vector or cross product
4. Coplanar _____ is linear mixture B and C
5. Discover the sine of the angle between the vectors, p = I + j + ok and q = 8i + 2j + 3k.
6. Three factors A, B and C exist with

KEY WORDS

• VECTORS
• SCALAR
• DOT PRODUCT
• DIMENSION
• PERPENDICULAR
• COLLINEAR
• COPLANAR
• UNIT VECTOR
• DETERMINANT