0

Guest

Courses New

Can the magnitude of the difference between two vectors ever be greater than the magnitude of either vector? Can it be greater than the magnitude of their sum? Give examples.

Can the magnitude of the difference between two vectors ever be greater than the magnitude of either vector? Can it be greater than the magnitude of their sum? Give examples.

Grade:upto college level

1 Answers

Deepak Patra
askIITians Faculty 471 Points
9 years ago
Consider a two dimensional vector\overrightarrow{a} with componenta_{x}} along the unit vector in x-direction a_{y}, along the unit vector in y-direction respectively.
Mathematically, the vector\overrightarrow{a} can be represented as:
\overrightarrow{a} = a_{x}\widehat{i}+ a_{y}\widehat{j}
The magnitude of the vector \overrightarrow{a} is given as:
|\overrightarrow{a}| = \sqrt{a_{x}^{2}+a_{}y^{2}}
Consider another vector\overrightarrow{b} with componentb_{x} along the unit vector in x-direction, b_{y} along the unit vector in y-direction respectively.
Mathematically, the vector\overrightarrow{b} can be represented as:
\overrightarrow{b} = b_{x}\widehat{i} + b_{y}\widehat{j}
The magnitude of the vector\overrightarrow{b} is given as:
|\overrightarrow{b}| = \sqrt{b_{x}^{2}+ \sqrt{b_{y}^{2}}
The difference of vectors\overrightarrow{a} and\overrightarrow{b} is given as:
\overrightarrow{r} = \overrightarrow{a} - \overrightarrow{b}
\overrightarrow{r} = (a_{x}\widehat{i} + a_{y}\widehat{j}) - (b_{x}\widehat{i}+ b_{y}\widehat{j})
\overrightarrow{r} = (a_{x} - b_{x})\widehat{i} + (a_{y}- b_{y})\widehat{j}
The magnitude of the vector\overrightarrow{r} can be given as:
|\overrightarrow{r}| = \sqrt{(a_{x}- b_{x})^{2} + (a_{y}- b_{y})^{2}} …… (1)
The magnitude of difference between the two vector given in the equation above depends on the value of (a_{x}- b_{x})and (a_{y}- b_{y})respectively. In a situation, where the signa_{x} of is opposite to that with the sign ofb_{x} or the sign ofa_{y} is opposite to the sign of b_{y}, the magnitude of vector\overrightarrow{r} can become greater to the magnitude of either vector\overrightarrow{a} or vector \overrightarrow{b}
For example,
Let us assume that vector \overrightarrow{a} is given as:
\overrightarrow{a} = 5\widehat{i} + 6\widehat{j}
Therefore the magnitude of vector\overrightarrow{a} is:
|\overrightarrow{a}| = \sqrt{(5)^{2}+ (6)^{2}}
= \sqrt{61}
Let us assume that the vector \overrightarrow{b}is given as:
\overrightarrow{b} = -\widehat{i}-2\widehat{j}
The magnitude of vector \overrightarrow{b} is:
|\overrightarrow{b}| = \sqrt{(1)^2 + (2)^{2}}
= \sqrt{5}
Now we calculate the magnitude of vector\overrightarrow{r} , which is:
|\overrightarrow{r}| = \sqrt{(5-(-1))^{2} + (6-(-2))^{2}}
= \sqrt{(6)^{2} + (8)^{2}}
= \sqrt{100}
Therefore the magnitude of vector\overrightarrow{r} is greater than the magnitude of vector\overrightarrow{a} and vector\overrightarrow{b}.
The sum of vector \overrightarrow{a} and vector\overrightarrow{b} is given as:
\overrightarrow{s} = \overrightarrow{a} + \overrightarrow{b}
\overrightarrow{s} = (a_{x} + b_{x})\widehat{i} + (a_{y}+b_{y})\widehat{j}
The magnitude of the vector\overrightarrow{s} can be given as:
|\overrightarrow{s}| = \sqrt{(a_{x}+ b_{x})^{2} + (a_{y}+ b_{y})^{2}} …… (2)
If the magnitude of vector\overline{r} is greater than the magnitude of vector\overline{s} , we have
233-883_9.PNG
Therefore, if the vectors\overrightarrow{a} and\overrightarrow{b} are such that the conditiona_{x}b_{x} < -a_{y}b_{y} is fulfilled, the magnitude of vector\overrightarrow{r} will be greater than the magnitude of vector\overrightarrow{s} .
For example,
Let us assume that vector \overrightarrow{a} is given as:
\overrightarrow{a} = 5\widehat{i} - 6\widehat{j}
Let us assume that the vector\overrightarrow{b} is given as:
\overrightarrow{b} = \widehat{i} + 2\widehat{j}
Now, we check if the condition is fulfilled by substituting the components of vectors as:
a_{x} = 5
a_{y} = -6
b_{x} = 1
b_{y} = 2
Substituting the values in the condition, we have
(5) (1) <-(-6)(2)
5 <12
Thus, our condition holds for the given vectors.
Now, we calculate the magnitude of vector\overrightarrow{r} and vector\overrightarrow{s} as:
The magnitude of vector \overrightarrow{s} is:
233-2373_10.PNG
The magnitude of vector\overrightarrow{r} is:

233-74_11.PNG
Therefore the magnitude of vector\overrightarrow{r} is greater than the magnitude of vector\overrightarrow{s} , as we expected it to be.

Think You Can Provide A Better Answer ?

Other Related Questions on Mechanics

View all Questions »

ASK QUESTION

Get your questions answered by the expert for free

Answer and earn gift vouchers

Win Gift vouchers upto Rs 500/-

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Click Here Know More

Complete Self Study Package designed by Industry Leading Experts

Click Here Know More

Live 1-1 coding classes to unleash the Creator in your Child

Click Here Know More

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Click Here Know More