Class 11th

Vector resolution is important to break complex motion into simpler, one-dimensional components. With it, we can simplify vector addition and analysis in 2D or 3D motion. Generally, we rely on perpendicular directions. That makes vector operations with displacement, velocity, and acceleration more manageable and accurate.

Scalar multiplication requires one vector and one scalar. Dot product requires two vectors. That's the basic difference. 

Vectors are not just numbers like scalars that have only magnitude. Vectors have direction and magnitude, and multiplying two vectors or multiplying a vector by a real number would yield results as per rules of vector algebra. 

The three different ways of vector multiplication are

• Scalar Multiplication - This is multiplying a vector with a scalar quantity. This operation results in another vector. 
• Dot Product - This is an operation that tells us that two vectors gives a scalar as a result. 
• Cross Product - This is an operation, when two vectors are multiplied, yields a vector that is perpendicular to both. 

Newton's First Law is important because it explains the basic concept of a body in motion. The first law implies that an object does not change the state of uniform motion or its state of rest unless a force acts on them. It helps us understand why seat belts, brakes, and handrails are needed. It tells us that a body resists changes already in motion due to inertia, which could lead to accidents without support.

In alkaline earth metals cationic size increases down the groups, hence thermal stability of carbonates increases down the group. Therefore both assertion and reason are correct and reason is correct explanation of assertion.

Phase difference between the two waves,

$?=k{x}_0=6.28*3.5=7\pi rad$              

Amplitude of resulting wave,

$A=\sqrt[]{A_1^2+A_2^2+2A_1{A}_{2}cos?}$              

$=\sqrt[]{12^2+5^2+2*12*5*cos\left(7\pi \right)}$              

$kx=\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

$\Rightarrow 1.57x=\frac{\pi }{2}=\frac{3.14}{2}=1.57$

x = 1 cm