Mathematics Exam > Mathematics Videos > Algebra > Properties of Scalar Multiplication
Properties of Scalar Multiplication Video Lecture | Algebra - Mathematics
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FAQs on Properties of Scalar Multiplication Video Lecture - Algebra - Mathematics
| 1. What is scalar multiplication? |  |
Ans. Scalar multiplication is an operation where a scalar (a real number) is multiplied to each element of a given vector. This operation results in a new vector with the same direction as the original vector but with a different magnitude.
| 2. How is scalar multiplication performed? | |
Ans. Scalar multiplication is performed by multiplying each component of the vector by the scalar. For example, if we have a vector [2, 4] and we want to multiply it by a scalar 3, the resulting vector would be [6, 12].
| 3. What are the properties of scalar multiplication? | |
Ans. The properties of scalar multiplication are as follows:
- Associativity: (a * b) * v = a * (b * v), where a and b are scalars and v is a vector.
- Distributivity with respect to addition: (a + b) * v = a * v + b * v, where a and b are scalars and v is a vector.
- Distributivity with respect to scalar addition: a * (v + w) = a * v + a * w, where a is a scalar and v, w are vectors.
- Identity element: 1 * v = v, where 1 is the multiplicative identity and v is a vector.
| 4. How does scalar multiplication affect the magnitude of a vector? | |
Ans. Scalar multiplication affects the magnitude of a vector by scaling it. If the scalar is greater than 1, the magnitude of the resulting vector will be greater than the magnitude of the original vector. If the scalar is between 0 and 1, the magnitude of the resulting vector will be smaller than the magnitude of the original vector. If the scalar is negative, the resulting vector will have the same magnitude as the original vector but will point in the opposite direction.
| 5. Can scalar multiplication change the direction of a vector? | |
Ans. No, scalar multiplication does not change the direction of a vector. It only scales the vector by multiplying it with a scalar. The resulting vector will have the same direction as the original vector, but its magnitude will be different depending on the scalar value.
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Dec 23, 2024 Last updated |
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