No. Cos theta (Cos θ) is a trigonometric function.
A vector is any physical quantity which has both magnitude and direction.
For example, Displacement. Displacement has a magnitude like 240m or 0 or 13 m, etc. It also depends on the direction. If an object moves along the positive direction of x-axis, then the displacement will have a positive sign and if it moves along the negative direction of x-axis, then displacement will be negative. Thus, it has both direction and magnitude and so is a vector.
Cos theta is a trigonometric function, strictly speaking.
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tan theta = sqrt(2)/2 = 1/sqrt(2).
Yes, a vector can be represented in terms of a unit vector which is in the same direction as the vector. it will be the unit vector in the direction of the vector times the magnitude of the vector.
The zero vector is both parallel and perpendicular to any other vector. V.0 = 0 means zero vector is perpendicular to V and Vx0 = 0 means zero vector is parallel to V.
Resultant vector or effective vector
Vector spaces can be formed of vector subspaces.