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 Ê.
Step 1: Find the magnitude of Ē
The magnitude of a vector Ē = xi + yj is given by the formula: |Ē| = √(x^2 + y^2)
Given Ē = 12ˆi + 5ˆj, we can calculate its magnitude as follows:
|Ē| = √(12^2 + 5^2)
= √(144 + 25)
= √(169)
= 13
Step 2: Find the unit vector Ê
The unit vector Ê in the direction of Ē can be obtained by dividing Ē by its magnitude: Ê = Ē/|Ē|
Given |Ē| = 13, Ê = (12ˆi + 5ˆj)/13
= 12/13ˆi + 5/13ˆj
Now that we have Ê, we can calculate the dot product Ā·Ê.
Step 3: Calculate the dot product Ā·Ê
The dot product of two vectors A = aˆi + bˆj and B = cˆi + dˆj is given by the formula: A·B = ac + bd
Given Ā = -4ˆi + 3ˆj and Ê = 12/13ˆi + 5/13ˆj, we can calculate Ā·Ê as follows:
Ā·Ê = (-4)(12/13) + (3)(5/13)
= -48/13 + 15/13
= -33/13
To plot Ā and Ê, we can use a graphing tool or software. On a Cartesian coordinate system, plot Ā using the coordinates (-4, 3) and Ê using the coordinates (12/13, 5/13) as the direction vector. The dot product Ā·Ê can be visualized as the projection of Ā onto Ê, which can be represented as a line segment from the origin to the point on the line defined by Ê that intersects with Ā when extended.