How is the product of a complex number and a real number represented on the coordinate plane?
A)When 3 + 2i is multiplied by 3, the result is 9 + 6i. Graphically, this shows that the product is a scalar and a counterclockwise rotation of 90° of the complex number.
B)When 3 + 2i is multiplied by 3, the result is 9 + 6i. Graphically, this shows that the product is a counterclockwise rotation of 90° of the complex number.
C)When 3 + 2i is multiplied by 3, the result is 9 + 6i. Graphically, this shows that the product is a scalar of the complex number.
D)When 3 + 2i is multiplied by 3, the result is 9 + 6i. Graphically, this shows that the product is a scalar and a clockwise rotation of 90° of the complex number.
B) When 3 + 2i is multiplied by 3, the result is 9 + 6i. Graphically, this shows that the product is a counterclockwise rotation of 90° of the complex number.
C) When 3 + 2i is multiplied by 3, the result is 9 + 6i. Graphically, this shows that the product is a scalar of the complex number.
To understand how the product of a complex number and a real number is represented on the coordinate plane, we need to consider the properties of complex numbers. In the complex plane, a complex number is represented by a point with coordinates (a, b), where 'a' represents the real part and 'b' represents the imaginary part.
When a complex number is multiplied by a real number, we simply multiply the real and imaginary parts of the complex number by the real number. In this case, we have (3 + 2i) * 3 = 9 + 6i. This means that we are multiplying 3 by both the real part (3) and the imaginary part (2i) of the complex number.
Geometrically, this multiplication by a real number scales the complex number by the magnitude of the real number. The direction of the complex number does not change. In this case, the real number 3 scales the complex number (3 + 2i) by a factor of 3, giving us a new complex number (9 + 6i). This shows that the product is a scalar (a scaling factor) of the original complex number.
Therefore, option C) is the correct answer: When 3 + 2i is multiplied by 3, the result is 9 + 6i. Graphically, this shows that the product is a scalar of the complex number.
multiplying a complex number by a scalar does not affect the angle in the Argand plane.
Thus no rotation is involved, it merely affects the magnitude.
e.g. 3 + 2i
tanØ = 2/3
3(3 + 2i) = 9 + 6i
tanØ = 6/9 = 2/3 , angle did not change
magnitude of 3 + 2i = √(9+4) = √13
magnitude of 9 + 6i = √(81+36) = √117 = √9*√13 = 3√13 <--- 3 times as long