The force determined by the vector r=(1,-8,-7) is resolved along three mutually perpendicular directions, one of which is in the direction of the vector a=2i+2j+k. Then the vector component of the force r along vector a is
To find the vector component of the force vector r along vector a, we need to use the formula:
Component of r along a = (r · a / |a|²) * a
where r · a is the dot product of vectors r and a, and |a|² is the magnitude squared of vector a.
First, we need to calculate the dot product of vectors r and a:
r · a = (1)(2) + (-8)(2) + (-7)(1)
= 2 - 16 - 7
= -21
Next, we need to calculate the magnitude squared of vector a:
|a|² = (2)² + (2)² + (1)²
= 4 + 4 + 1
= 9
Now, substituting the values into the formula, we have:
Component of r along a = (-21 / 9) * (2i + 2j + k)
= (-7 / 3) * (2i + 2j + k)
= (-14/3)i + (-14/3)j + (-7/3)k
Therefore, the vector component of the force vector r along vector a is (-14/3)i + (-14/3)j + (-7/3)k.
To find the vector component of the force vector r along vector a, you can use the dot product. The dot product of two vectors A and B is given by the formula:
A · B = |A| |B| cosθ
Where |A| and |B| are the magnitudes of vectors A and B respectively, and θ is the angle between the two vectors.
In this case, we want to find the component of vector r along vector a, so we can use the dot product as follows:
r · a = |r| |a| cosθ
First, let's calculate the magnitude of vector a:
|a| = sqrt((2^2) + (2^2) + (1^2)) = sqrt(4 + 4 + 1) = sqrt(9) = 3
Next, let's calculate the dot product of vectors r and a:
r · a = (1)(2) + (-8)(2) + (-7)(1) = 2 - 16 - 7 = -21
Finally, let's calculate the magnitude of vector r:
|r| = sqrt((1^2) + (-8^2) + (-7^2)) = sqrt(1 + 64 + 49) = sqrt(114) ≈ 10.68
Now, we can calculate the vector component of r along a:
Component of r along a = (r · a) / |a|
Component of r along a = -21 / 3 = -7
Therefore, the vector component of the force r along vector a is -7.
r•a/|a|
Since we know that,
Projection of r vector along a vector=
(r vector)•(a^)
=(i-8j-7k)•(2/3xi+2/3xj+1/3xk)
=(2/3xi-16/3xj-7/3xk)