Given vector A= 3 i cap + j cap + 4k cap and vector B = i cap + 3 j cap - 5 k cap. Find the unit vector in the direction of (vector A+ vector B) and (vector A - vector B)
A+B=4i+4j-k. Find the magnitude of A+B: mag(A+B)=sqrt(16+16+1)=sqrt(33). So the unit vector of A+B is (1/sqrt(33))*(4i+4j-k). A-B=2i-2j+9k. mag(A-B)=sqrt(4+4+81)=sqrt(89).The unit vector of A-B is (1/sqrt(89))*(2i-2j+9k)
To find the unit vector in the direction of a given vector, we need to divide the vector by its magnitude.
Let's start by finding the vector sum of A and B: A + B.
A + B = (3 i + j + 4k) + (i + 3j - 5k)
= (3i + i) + (j + 3j) + (4k - 5k)
= 4i + 4j - k
Next, let's find the vector difference of A and B: A - B.
A - B = (3 i + j + 4k) - (i + 3j - 5k)
= (3i - i) + (j - 3j) + (4k + 5k)
= 2i - 2j + 9k
Now, let's calculate the magnitudes of A + B and A - B.
Magnitude of A + B = √((4)^2 + (4)^2 + (-1)^2)
= √(16 + 16 + 1)
= √33
Magnitude of A - B = √((2)^2 + (-2)^2 + (9)^2)
= √(4 + 4 + 81)
= √89
To find the unit vector in the direction of A + B, divide the vector (4i + 4j - k) by its magnitude √33:
Unit vector in the direction of A + B = (4i + 4j - k) / √33
To find the unit vector in the direction of A - B, divide the vector (2i - 2j + 9k) by its magnitude √89:
Unit vector in the direction of A - B = (2i - 2j + 9k) / √89
These are the unit vectors in the direction of A + B and A - B, respectively.