Theorem
Statement
If and be any three non-zero and non coplanar space vectors then any other vector can be expressed uniquely as the sum of three space vectors parallel to the vectors and .
Proof: Let , and be any three non-zero and non coplanar space vectors as shown in the figure. Let be any other vector in the space. Join OE, OC and AC. Then by the parallelogram law of the addition of the vectors, we get = . Again . Hence = or ..................(i)
For uniqueness, if possible,let ...............................(ii). Equating (i) and (ii) we get x =x ' , y = y' and z = z'. Hence the expression (i) is unique. Hence the theorem.
Note: See the related figure in the next resource page.
New Resources
- Base X Machine (Addition with input)
- Alternate Segment Theorem
- Point(s), a Line, Ray(s), Segment(s), & an angle - interactive & 3 questions
- Special Line through Triangle V1 (Theorem Discovery)
- Six Pattern Blocks (14 each)
- Language: English
Statement
If and be any three non-zero and non coplanar space vectors then any other vector can be expressed uniquely as the sum of three space vectors parallel to the vectors and .
Proof: Let , and be any three non-zero and non coplanar space vectors as shown in the figure. Let be any other vector in the space. Join OE, OC and AC. Then by the parallelogram law of the addition of the vectors, we get = . Again . Hence = or ..................(i)
For uniqueness, if possible,let ...............................(ii). Equating (i) and (ii) we get x =x ' , y = y' and z = z'. Hence the expression (i) is unique. Hence the theorem.
New Resources
- Base X Machine (Addition with input)
- Alternate Segment Theorem
- Point(s), a Line, Ray(s), Segment(s), & an angle - interactive & 3 questions
- Special Line through Triangle V1 (Theorem Discovery)
- Six Pattern Blocks (14 each)
- Language: English