Question
$\text{If}\ \ \vec{\text{a}}=\vec{\text{b}}+\vec{\text{c}},$ then is it true that $\big|\vec{\text{a}}\big|=\big|\vec{\text{b}}\big|+\big|\vec{\text{c}}\big|?$ Justify your answer.
✓
Answer
$\text{If}\ \triangle\text{ABC},\ \text{let}\ \overrightarrow{\text{CB}}=\vec{\text{a}},\ \overrightarrow{\text{CA}}=\vec{\text{b}},\text{and}\ \overrightarrow{\text{AB}}=\vec{\text{c}}$ (as shown in the following figure).
Now, by the triangle law of vector additio, we have $\vec{\text{a}}=\vec{\text{b}}+\vec{\text{c}}$It is clearly known that $|\vec{\text{a}}|,\big|\vec{\text{b}}\big|,\ \text{and}\ \big|\vec{\text{c}}\big|$ represent the sides of $\triangle\text{ABC}.$
Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side. $\therefore\ |\vec{\text{a}}|<\big|\vec{b}\big|+|\vec{c}|$ Hence, it is not true that $|\vec{\text{a}}|=\big|\vec{\text{b}}\big|+\big|\vec{\text{c}}\big|$
Now, by the triangle law of vector additio, we have $\vec{\text{a}}=\vec{\text{b}}+\vec{\text{c}}$It is clearly known that $|\vec{\text{a}}|,\big|\vec{\text{b}}\big|,\ \text{and}\ \big|\vec{\text{c}}\big|$ represent the sides of $\triangle\text{ABC}.$
Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side. $\therefore\ |\vec{\text{a}}|<\big|\vec{b}\big|+|\vec{c}|$ Hence, it is not true that $|\vec{\text{a}}|=\big|\vec{\text{b}}\big|+\big|\vec{\text{c}}\big|$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the vector along vector $5 \hat{i}-\hat{j}+2 \hat{k}$ whose magnitude is 8 units.
→If $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2+\big|\vec{\text{a}}.\vec{\text{b}}\big|^2=400$ and $|\vec{\text{a}}|=5,$ then write the value of $\big|\vec{\text{b}}\big|.$
→If $\text{A}=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix},$ find $x$ satisfying $0<\text{x}<\frac{\pi}{2}$ when $A + A^T = I$
→Four cards are drawn simultaneously from a well shuffled pack of 52 playing cards. Find the probability distribution of the number of aces.
Evaluate the following integrals:
$\int\frac{\text{x}\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}\text{dx}$
→$\int\frac{\text{x}\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}\text{dx}$
Evaluate the following integrals:
$\int\frac{\sec^2\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$
→$\int\frac{\sec^2\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$
Solve the following differential equations:$(1-\text{x}^2)\text{dy + xy dx = xy}^2\text{ dx}$
→Write the value of $\sin^{-1}(\sin(-600^\circ))\sin(-600^\circ).$
→An urn contain 5 red and 2 black balls. Two balls rendomly selected. Let X represent the number of black ball. What are the possible values of X. Is X a random variable?
Evaluate: $\int \frac{x^2+1}{\left(x^2+2\right)\left(2 x^2+1\right)} d x$
→