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Homework Statement
i have to find the scalar and vector projection of a=i-j+k and b=2i-j-2k
and i got:
Vector proj = (1/3)(i-j+k) = i/3 + j/3 + k/3
scalar proj = (1/9)(2i-j-2k) = 2i/9 - j/9 - 2k/9
is this correct?
A vector is a quantity that has both magnitude and direction, while a scalar is a quantity that only has magnitude. In other words, a vector describes both how much and in what direction something is moving, while a scalar only describes how much.
The vector projection of a onto b is calculated by taking the dot product of a and the unit vector in the direction of b. This can be written as a · (b/|b|), where |b| represents the magnitude of b. The result is a vector that points in the same direction as b, with a magnitude proportional to the component of a in that direction.
The scalar projection of a onto b is the magnitude of the vector projection of a onto b. In other words, it is the length of the shadow that a casts onto the direction of b.
The scalar projection of a onto b can be calculated by dividing the dot product of a and b by the magnitude of b. This can be written as (a · b)/|b|.
Yes, the scalar projection of a onto b can be negative if the angle between a and b is greater than 90 degrees. This indicates that the component of a in the direction of b is in the opposite direction of b's magnitude, resulting in a negative value for the scalar projection.