Calculate the dot product Ā·Ê(where Ê = Ē/E ~ is the unit vector in the direction of Ē ) and plot both Ā and Ê, showing how Ā·Ê is the projection of Ā on Ê.
Ā= −4ˆi + 3ˆj
Ē= 12ˆi + 5ˆj
To calculate the dot product Ā·Ê, we first need to find the unit vector in the direction of Ē.
1. Find the magnitude of Ē: The magnitude of a vector Ē is given by ∥Ē∥ = √(E_x^2 + E_y^2), where E_x and E_y are the components of Ē. For Ē = 12ˆi + 5ˆj, we have ∥Ē∥ = √(12^2 + 5^2) = √(144 + 25) = √169 = 13.
2. Calculate the unit vector Ê: The unit vector Ê in the direction of Ē is given by Ê = (Ē / ∥Ē∥), where Ē is the vector and ∥Ē∥ is its magnitude. For Ē = 12ˆi + 5ˆj, we have Ê = (12ˆi + 5ˆj) / 13.
Now that we have Ċiuml ; = ∑ ( Ā_i x Ê_i ) = ( Ā_x x Ê_x ) + ( Ā_y x Ê_y )
we can substitute the values for Ā and Ê:
To find Ā·Ê, we can use the formula:
Ā·Ê = (Ā_x * Ê_x) + (Ā_y * Ê_y)
3. Calculate Ā·Ê: For Ā = -4ˆi + 3ˆj and Ê = (12ˆi + 5ˆj) / 13, we have:
Ā·Ê = (-4 * (12/13)) + (3 * (5/13))
= (-48/13) + (15/13)
= -33/13
So, Ā·Ê = -33/13.
To plot both Ā and Ê and show how Ā·Ê is the projection of Ā on Ê, follow these steps:
4. Draw a coordinate plane: Draw a 2D Cartesian coordinate plane.
5. Plot Ā: Plot the vector Ā = -4ˆi + 3ˆj on the coordinate plane. Start from the origin (0, 0) and move -4 units in the x-direction and 3 units in the y-direction.
6. Plot Ê: Plot the vector Ê = (12ˆi + 5ˆj) / 13 on the coordinate plane. Start from the origin (0, 0) and move 12/13 units in the x-direction and 5/13 units in the y-direction.
7. Draw the projection: Draw a line segment from the origin to the endpoint of Ā·Ê. This represents the projection of Ā onto Ê.
By following these steps, you will have calculated the dot product Ā·Ê and also plotted both Ā and Ê, showing the projection of Ā on Ê.