can someone please help me with this question?
A vector has components x =4 , y =2. When the vector is multiplied by the scalar 9, how does its magnitude and direction change?
the direction does not change
the x and y values both are multiplied by 9
k(ai + bj) = kai + kbj
Certainly! To understand how the magnitude and direction of a vector change when it is multiplied by a scalar, let's break it down.
First, let's find the initial magnitude (or length) of the vector. The magnitude of a vector with components x and y can be calculated using the Pythagorean theorem:
Initial Magnitude (M₁) = √(x₁² + y₁²)
In this case, x₁ = 4 and y₁ = 2:
Initial Magnitude (M₁) = √(4² + 2²) = √(16 + 4) = √20 = 2√5
Now, when the vector is multiplied by a scalar 9, each component is multiplied by that scalar:
x₂ = 4 * 9 = 36
y₂ = 2 * 9 = 18
Next, let's find the magnitude of the new vector after the scalar multiplication. Using the same formula as before:
New Magnitude (M₂) = √(x₂² + y₂²)
In this case, x₂ = 36 and y₂ = 18:
New Magnitude (M₂) = √(36² + 18²) = √(1296 + 324) = √1620 = 6√10
To understand how the magnitude has changed, we can compare the initial and new magnitudes:
M₂ / M₁ = (6√10) / (2√5) = (6/2) * (√10 / √5) = 3 * (√(10/5)) = 3 * (√2) = 3√2
Therefore, the magnitude of the vector increases by a factor of 3√2 when multiplied by the scalar 9.
Regarding the direction, multiplying a vector by a scalar does not affect the direction, only the magnitude changes. In this case, the direction of the vector remains the same.
In summary, when the vector with components x = 4 and y = 2 is multiplied by the scalar 9, its magnitude increases by a factor of 3√2, while its direction remains unchanged.