Find the vector v with the given magnitude and the same direction as u. ||v||=7, u=3i+4j
a unit vector in the direction of u is (3/5)i + (4/5)j
so a vector with length 7 with that direction is
(21/5)i + 28/5)j
check: magnitude = √((21/5)^2 + (28/5))
= √(1225/25) = √49 = 7
Well, it sounds like we need to find a vector with the same direction as u but with a magnitude of 7. It's very important for vectors to maintain their sense of direction, just like knowing where you're going when you're driving.
Let's start by finding the unit vector of u. The unit vector is a vector with a magnitude of 1, but with the same direction as u. So, we divide vector u by its magnitude:
û = (3/5)i + (4/5)j.
Now that we have the unit vector, we can simply multiply it by 7 to get a vector with the same direction as u but with a magnitude of 7:
v = 7û = 7((3/5)i + (4/5)j) = (21/5)i + (28/5)j.
So, v is equal to (21/5)i + (28/5)j. But remember, v not only has the same direction as u, but also a magnitude of 7. Now that's what I call a vector comedy duo!
To find a vector v with the given magnitude and the same direction as u, we can use the equation v = (||v||/||u||) * u, where ||v|| is the magnitude of vector v, ||u|| is the magnitude of vector u, and u is the given vector.
Given that ||v|| = 7 and u = 3i + 4j, let's calculate v.
First, we need to find the magnitude of u.
||u|| = sqrt( (3^2) + (4^2) ) = sqrt(9 + 16) = sqrt(25) = 5.
Now, we can substitute the values into the equation to calculate v.
v = (||v||/||u||) * u = (7/5) * (3i + 4j) = (21/5)i + (28/5)j.
Therefore, the vector v with the given magnitude and the same direction as u is v = (21/5)i + (28/5)j.
To find the vector v with the same direction as u and a magnitude of 7, we will use the concept of unit vectors.
The unit vector in the same direction as u is obtained by dividing the vector u by its magnitude.
In this case, the magnitude of u is given as √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Now, the unit vector in the same direction as u is given by:
unit vector u = u / ||u||
Substituting the values, we have:
(unit vector u) = (3i + 4j) / 5
To obtain a vector of magnitude 7 in the same direction as u, we multiply the unit vector by 7.
Therefore, the vector v = 7 * (unit vector u)
= 7 * ((3i + 4j) / 5)
= (21/5)i + (28/5)j
Thus, the vector v with a magnitude of 7 and the same direction as u is (21/5)i + (28/5)j.