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mathsVector AlgebraMultiplication of Vectors by Scalar

### Repeated addition of a Vector

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We have learned about addition of two or more vectors. Consider a vector being repeatedly added to itself. What is the result of the addition vec p + vec p?

• sum = 2 times vec p
• sum = 2 times vec p
• component form is required to find the sum
• directional cosines are required to find the sum
• sum =2 times |vec p|

The answer is 'sum = 2 times vec p'. If vec p = ai+bj+ck then vec p + vec p = 2ai+2bj+2ck. This equals 2 times vec p.

Repeated addition can be generalized to multiplication of vector by a scalar. Scalar multiplier is denoted with lambda to identify differently to the component values a, b, c. Note that the scalar multiplier and component values are real numbers. What is the result of lambda vec p if vec p = ai+bj+ck?

• lambda ai+bj+ck
• lambda i+ lambda j + lambda k
• lambda ai+ lambda bj + lambda ck
• lambda ai+ lambda bj + lambda ck
• lambda sqrt(a^2+b^2+c^2)

The answer is 'lambda a i+ lambda b j + lambda c k'.

When vector is multiplied by a scalar, the vector scales up or down proportionally.

Multiplication of Vector by a scalar: For any vector vec p = ai + bj+ ck in bbb V and scalar lambda in RR
lambda vec p= lambda ai+ lambda bj + lambda ck

Can a vector vec p be divided by a real number lambda?

• Yes. Division is inverse of multiplication.
• Yes. Division is inverse of multiplication.
• No. Division of vector is not defined.

The answer is 'Yes. Division is inverse of multiplication.'

If the given vector is vec p = ai+bj+ck then
vec p -: lambda
quad quad = 1/lambda xx vec p
`quad quad = a/lambda i+b/lambda j+c/lambda k

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