Question

If $A, B, C, D$ are four points and $\vec{AB} = \vec{DC}$ , then $\vec{AC} + \vec{BD} = $

• A. $2 \vec{AD}$
• B. $2 \vec{CB}$
• C. $2 \vec{AC}$
• D. None of these

Answer 1

The Correct Option is A

Since , \(\overline{AB} = \overline{DC}\)
\(\therefore \:\:\: \overline{OB} - \overline{OA} = \overline{OC} - \overline{OD}\)
\(\Rightarrow \:\: \overline{OC} - \overline{OB} = \overline{OD} - \overline{OA}\) ....(i) 
Now, \(\overline{AC} - \overline{BD} = \overline{OC} - \overline{OA} + \overline{OD} - \overline{OB}\)
\(= (\overline{OC} - \overline{OB})+( \overline{OD} - \overline{OA} )\)
\(= (\overline{OD} - \overline{OA})+( \overline{OD} - \overline{OA} )\) (Using (i)) 
\(= 2(\overline{OD} - \overline{OA} ) = 2 \overline{AD}\)

A vector is a unit of measure that lacks a location but has both magnitude and direction. Examples include speed and acceleration. A vector is a directed line segment with an arrow indicating its direction and a length equal to the vector's magnitude. The vector is often represented by an arrow whose length is proportional to the magnitude of the quantity and whose direction is comparable to that of the quantity. Additionally, scaled vector diagrams are typically used to depict vector quantities. The displacement vector is represented in the vector diagram.

As was already established, the vector is essentially represented by an arrow, whose length is directly proportional to the magnitude of the quantity and whose direction is the same as the quantity. The displacement vector will be shown in the vector diagram:

Vectors are the name given to the mathematical representation of physical quantities that include both the magnitude and the direction.

Vectors must have their resultant independently determined since they are added geometrically rather than algebraically.

This vector's magnitude is expressed as either |ab| or |a|. With the aid of the Pythagorean theorem, it illustrates the vector length and shows how every physical quantity's vector may be represented as a straight line with an arrowhead. The length of the straight line indicates the vector's magnitude, and the arrowhead indicates its direction. 

The diagonal of a parallelogram formed from the same point will describe the outcome if the two forces are represented as vectors a and b by the neighboring sides of the parallelogram. Vector (r = a + b) is the rule of addition for parallelograms.