Given $a+b+c+d=0,$ which of the following statements eare correct:
$(a)\;a, b,$ c, and $d$ must each be a null vector,
$(b)$ The magnitude of $(a+c)$ equals the magnitude of $(b+d)$
$(c)$ The magnitude of a can never be greater than the sum of the magnitudes of $b , c ,$ and $d$
$(d)$ $b + c$ must lie in the plane of $a$ and $d$ if $a$ and $d$ are not collinear, and in the line of a and $d ,$ if they are collinear ?
Given $a+b+c+d=0,$ which of the following statements eare correct:
$(a)\;a, b,$ c, and $d$ must each be a null vector,
$(b)$ The magnitude of $(a+c)$ equals the magnitude of $(b+d)$
$(c)$ The magnitude of a can never be greater than the sum of the magnitudes of $b , c ,$ and $d$
$(d)$ $b + c$ must lie in the plane of $a$ and $d$ if $a$ and $d$ are not collinear, and in the line of a and $d ,$ if they are collinear ?
$(a)$ Incorrect : In order to make $a+b+c+d=0,$ it is not necessary to have all the four given vectors to be null vectors. There are many other combinations which can give the sum zero.
$(b)$ Correct : $a + b + c + d = 0 a + c =-( b + d )$
Taking modulus on both the sides, we get:
$| a + c |=|-( b + d )|=| b + d |$
Hence, the magnitude of $(a+c)$ is the same as the magnitude of $(b+d)$
$(c)$ Correct : $a+b+c+d=0 a=(b+c+d)$
Taking modulus both sides, we get:
$| a |=| b + c + d |$
$| a | \leq| a |+| b |+| c | \ldots \ldots(i)$
Equation $(i)$ shows that the magnitude of $a$ is equal to or less than the sum of the magnitudes of $b , c ,$ and $d$ Hence, the magnitude of vector $a$ can never be greater than the sum of the magnitudes of $b , c ,$ and $d$
$(d)$ Correct : For $a+b+c+d=0$
The resultant sum of the three vectors $a,(b+c),$ and $d$ can be zero only if $(b+c)$ lie in a plane containing a and $d$, assuming that these three vectors are represented by the three sides of a triangle.
If $a$ and $d$ are collinear, then it implies that the vector ( $b+c$ ) is in the line of $a$ and $d$. This implication holds only then the vector sum of all the vectors will be zero.
Similar Questions
Two forces $P$ and $Q$, of magnitude $2F$ and $3F$, respectively, are at an angle $\theta $ with each other. If the force $Q$ is doubled, then their resultant also gets doubled. Then, the angle $\theta $ is ....... $^o$
Two forces $P$ and $Q$, of magnitude $2F$ and $3F$, respectively, are at an angle $\theta $ with each other. If the force $Q$ is doubled, then their resultant also gets doubled. Then, the angle $\theta $ is ....... $^o$
- [JEE MAIN 2019]
Let $\overrightarrow C = \overrightarrow A + \overrightarrow B$
$(A)$ It is possible to have $| \overrightarrow C | < | \overrightarrow A |$ and $ | \overrightarrow C | < | \overrightarrow B|$
$(B)$ $|\overrightarrow C |$ is always greater than $|\overrightarrow A |$
$(C)$ $|\overrightarrow C |$ may be equal to $|\overrightarrow A | + |\overrightarrow B|$
$(D)$ $|\overrightarrow C |$ is never equal to $|\overrightarrow A | + |\overrightarrow B|$
Which of the above is correct
Let $\overrightarrow C = \overrightarrow A + \overrightarrow B$
$(A)$ It is possible to have $| \overrightarrow C | < | \overrightarrow A |$ and $ | \overrightarrow C | < | \overrightarrow B|$
$(B)$ $|\overrightarrow C |$ is always greater than $|\overrightarrow A |$
$(C)$ $|\overrightarrow C |$ may be equal to $|\overrightarrow A | + |\overrightarrow B|$
$(D)$ $|\overrightarrow C |$ is never equal to $|\overrightarrow A | + |\overrightarrow B|$
Which of the above is correct
Two forces, ${F_1}$ and ${F_2}$ are acting on a body. One force is double that of the other force and the resultant is equal to the greater force. Then the angle between the two forces is
Two forces, ${F_1}$ and ${F_2}$ are acting on a body. One force is double that of the other force and the resultant is equal to the greater force. Then the angle between the two forces is
If the angle between $\hat a$ and $\hat b$ is $60^o$, then which of the following vector $(s)$ have magnitude one
$(A)$ $\frac{\hat a + \hat b}{\sqrt 3}$ $(B)$ $\hat a + \widehat b$ $(C)$ $\hat a$ $(D)$ $\hat b$
If the angle between $\hat a$ and $\hat b$ is $60^o$, then which of the following vector $(s)$ have magnitude one
$(A)$ $\frac{\hat a + \hat b}{\sqrt 3}$ $(B)$ $\hat a + \widehat b$ $(C)$ $\hat a$ $(D)$ $\hat b$
Explain the parallelogram method for vector addition. Also explain that this is comparable to triangle method.
Explain the parallelogram method for vector addition. Also explain that this is comparable to triangle method.