find a unit vector parallel to the resultant of vectors : A=2i^+j^-k^& B=3i^-2j^+2k^
well, the resultant
C = A+B = 5i-j+k
|C| = √27
so, the unit vector is C/√27
DER
To find a unit vector parallel to the resultant of vectors A and B, follow these steps:
Step 1: Find the resultant vector R.
To find the resultant vector, add vectors A and B component-wise.
R = A + B
= (2i + j - k) + (3i - 2j + 2k)
= (2i + 3i) + (j - 2j) + (-k + 2k)
= 5i - j + k
Step 2: Calculate the magnitude, ||R||, of the resultant vector.
The magnitude of a vector is calculated by taking the square root of the sum of squared components.
||R|| = sqrt((5^2) + (-1^2) + (1^2))
= sqrt(25 + 1 + 1)
= sqrt(27)
= 3√3
Step 3: Obtain the unit vector by dividing the resultant vector by its magnitude.
To obtain the unit vector, divide the resultant vector (R) by its magnitude (||R||).
Unit vector = R / ||R||
= (5i - j + k) / (3√3)
Therefore, a unit vector parallel to the resultant of vectors A and B is (5i - j + k) / (3√3).
To find a unit vector parallel to the resultant of two vectors A and B, follow these steps:
1. Start by calculating the resultant vector R by adding vectors A and B:
R = A + B
2. Determine the magnitude of the resultant vector R, denoted as |R|, using the formula:
|R| = √(Rx² + Ry² + Rz²)
Here, Rx, Ry, and Rz represent the x, y, and z components of the resultant vector R, respectively.
3. Normalize the resultant vector R to obtain a unit vector. Divide each component of the resultant vector R by its magnitude |R|:
R_unit = (Rx / |R|)i^ + (Ry / |R|)j^ + (Rz / |R|)k^
Here, i^, j^, and k^ represent the unit vectors along the x, y, and z axes, respectively.
Let's apply these steps to calculate a unit vector parallel to the resultant vector of A = 2i^ + j^ - k^ and B = 3i^ - 2j^ + 2k^.
Step 1: Calculate the resultant vector R.
R = A + B
= (2i^ + j^ - k^) + (3i^ - 2j^ + 2k^)
= (2 + 3)i^ + (1 - 2)j^ + (-1 + 2)k^
= 5i^ - j^ + k^
Step 2: Determine the magnitude |R|.
|R| = √(Rx² + Ry² + Rz²)
= √((5²) + (-1²) + (1²))
= √(25 + 1 + 1)
= √27
= 3√3
Step 3: Normalize the resultant vector R to get a unit vector.
R_unit = (Rx / |R|)i^ + (Ry / |R|)j^ + (Rz / |R|)k^
= (5 / (3√3))i^ + (-1 / (3√3))j^ + (1 / (3√3))k^
Therefore, a unit vector parallel to the resultant of vectors A and B is:
(5 / (3√3))i^ + (-1 / (3√3))j^ + (1 / (3√3))k^