If you intend 'dimensions' to mean units then whenever the two quantities are to be operated on each other then they must have the 'dimensions', refer to dimensional analysis
Two lines intersect at one point. If in two dimensions, and they do not intersect they are parallel. The other option in two dimensions is they are the co-linear, that is they are the same line, in which case they intersect at all points.
yes
They must be the same length.
The area of a rectangle is equal to its length times its width. So any two rectangles for which these dimensions have the same product, the area is the same. For example, a rectangle that is two meters wide and three meters long and one that is one meter wide and six meters long will both have an area of six square meters.
Perimeters must be the same
No
Yes.
Yes they must be in the same units of measurements.
No, it is not true.
Yes, two quantities must have the same dimensions if they are to be equated. This is because equations represent a balance or equality between the two quantities, and quantities with different dimensions cannot be compared or equated meaningfully. For example, equating speed (meters per second) to time (seconds) is invalid as they have different dimensions.
Yes, two quantities must have the same dimension in order to be added together. This is because addition involves combining like terms, and only quantities with the same dimensional units can be meaningfully combined. For example, you cannot add meters and seconds, as they represent different dimensions.
When two physical quantities are added together, they must have the same units in order to be combined. This means that both quantities must be expressed in the same type of measurement, such as meters or kilograms, for the addition to be valid.
Same direction and equal magnitudes.
The answer will depend on (a) whet the dimensions of the two quantities are, and (b) what the missing operator between the two quantities is.
Yes, always. One molecule plus one atom is not 2 of anything. One unit north plus one unit east is not 2 units northeast. ■
No. Measurement units are defined by and conversely. So the same units necessarily means same dimensions.
Yes, you can have two objects with identical masses by either having two objects made from the same material and dimensions or by adjusting the quantities of different materials to have the same mass.