why is dot product of two vectors a scalar quantity..?

and why is cross product of two vectors a vector quantity..?

a vector quantity can contain any sign: +ve or -ve. If we multiply a vector quantity with +ve sign with a vector quantity of -ve sign then the product will contain -ve sign.

thats why product of two vectors is not a scalar quantity. it is taken as vector.

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See, Vectors are quantities whose addition, multiplication etc. do not not follow the rules we use with real

numbers. Rather they are quantities which follow rules of complex numbers and matrix.

In daily life, we generally add two quanities having same unit/dimensions like adding rupees, connecting

batteries in series to add up their voltages. But two quantities having different units/dimensions cannot beadded like you cannot add 50 rupees with 50 volts to get 100 rupees. However sometimes, we need tomultiply two quantities where both of them may have same or different dimensions or sometimes one or bothof them may be dimensionless also. Like amount of work required in construction of some building iscalculated in terms of Man*hours, distance is calculated by multiplying speed*time. Now, when to add andwhen to multiply is a simple a question of mathematics and common sense.Above examples include only scalar quantities. However, in our daily life we see some of the quantities aresuch that their resultant is doesn't simply depend upon adding or multiplying their magnitudes but it alsodepends upon other factors (like direction) which are also characteristics of those two quantities only (not anyexternal entity). Take an example of the bicycle pedal. While pushing the pedals, when the cycle starts fromrest, you sit on the seat then first you have to bring the pedal bars in horizontal position by rotating it in reversedirection (where ratchet lets it free from load). Once it is set to horizontal condition, you start pushing it inforward direction and the cycle starts accelerating. Suppose the pedal were perfectly vertical when cycle wasat rest, no matter how hard you push the cycle will not move, if it is somewhere in between perfect vertical andperfect horizontal, the cycle will move. But you will find that the cycle gets maximum acceleration when thepedal comes in perfectly horizontal position. Also you will observe that for the same horizontal position, whenyou put your whole weight on one pedal, the cycle gets even higher acceleration. That means acceleration ofthe cycle depends on two factors: 1. Amount of force 2. angular position of pedal with respect to the direction ofapplication of force (which is usually vertically downwards when u push the pedal). If u carefully observe theamount of acceleration also depends upon how far pedals are situated from the axis of chain-wheel, this canbe observed by trying different size cycles. Hence third parameter is distance of pedal from axis of rotation.Thus we see that acceleration of cycle is simply not proportional toForce*Distance of pedal from axis, but also on angular position of pedal bars relative to force direction. Suchquantities are vector and you cannot get the resultant by simply multiplying their magnitudes. Results for suchquantities have been obtained by careful experiments and checking dependencies.Now, coming to dot and cross product. As you know vectors can be added using law of parallelogram wherethe diagonal is equal in magnitude as well as direction to the resultant vector. Similarly, if we have to multiplythe vectors and their result is also known to be a vector quantity, we need to find the area of the parallelogrammade by them. Please notice that area is a vector quantity. Now, area of parallelogram is equal tobase*height. Base will be equal to magnitude of one of the vectors while height will be equal to component ofthe other vector which is perpendicular to the first vector and this will be equal to length of second vectormultiplied by sine theta. Hence, if result is a vector we perform cross product and take sin theta.While in case of dot product we do not calculate area. It is a product only, however this is obtained bymultiplying the component of first vector which is parallel to the second vector with the second vector. Hencecos theta comes into picture. As the result is a scalar quantity, whether u take component of first parallel tosecond or component of second parallel to first. In both cases you will get the same result.

why is dot product of two vectors a scalar quantity..? and why is cross product of two vectors a vector quantity..?

why is dot product of two vectors a scalar quantity..?

and why is cross product of two vectors a vector quantity..?