Let the angle between two nonzero vectors A and B be 120° and its resultant be c : 1. C must be equal to |A-B| 2. C must be less than |A-B| 3. C must be greater than |A-B| 4. C may be equal to |A-B|

2. C must be less than |A-B| Explanation: Here, we have three vector A, B and C. | A + B |^2= | A |^2+ | B |^2+2 A . B ... (i) | A - B |^2= | A |^2+ | B |^2-2 A . B ... (ii) Subtracting (i) from (ii), we get: | A + B |^2- | A - B |^2=4 A . B Using the resultant property C = A + B , we get: | C |^2- | A - B |^2=4 A . B | C |^2= | A - B |^2+4 A . B | C |^2= | A - B |^2+4 | A |. | B | 120^ Since cosine is negative in the second quadrant, C must be less than |A-B|.

Let the angle between two nonzero vectors A and B be 120° and its…

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Question

Let the angle between two nonzero vectors $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ be 120° and its resultant be $\vec{\text{c}}:$

C must be equal to $|\text{A}-\text{B}|$
C must be less than $|\text{A}-\text{B}|$
C must be greater than $|\text{A}-\text{B}|$
C may be equal to $|\text{A}-\text{B}|$

✓

Answer

C must be less than $|\text{A}-\text{B}|$

Explanation:

Here, we have three vector A, B and C.

$\big|\overrightarrow{\text{A}}+\overrightarrow{\text{B}}\big|^2=\big|\overrightarrow{\text{A}}\big|^2+\big|\overrightarrow{\text{B}}\big|^2+2\overrightarrow{\text{A}}.\overrightarrow{\text{B}} \ ...{\text{(i)}}$

$\big|\overrightarrow{\text{A}}-\overrightarrow{\text{B}}\big|^2=\big|\overrightarrow{\text{A}}\big|^2+\big|\overrightarrow{\text{B}}\big|^2-2\overrightarrow{\text{A}}.\overrightarrow{\text{B}} \ ...{\text{(ii)}}$

Subtracting (i) from (ii), we get:

$\big|\overrightarrow{\text{A}}+\overrightarrow{\text{B}}\big|^2-\big|\overrightarrow{\text{A}}-\overrightarrow{\text{B}}\big|^2=4\overrightarrow{\text{A}}.\overrightarrow{\text{B}}$

Using the resultant property $\overrightarrow{\text{C}}=\overrightarrow{\text{A}}+\overrightarrow{\text{B}},$ we get:

$\big|\overrightarrow{\text{C}}\big|^2-\big|\overrightarrow{\text{A}}-\overrightarrow{\text{B}}\big|^2=4\overrightarrow{\text{A}}.\overrightarrow{\text{B}}$

$\Rightarrow\big|\overrightarrow{\text{C}}\big|^2=\big|\overrightarrow{\text{A}}-\overrightarrow{\text{B}}\big|^2+4\overrightarrow{\text{A}}.\overrightarrow{\text{B}}$

$\Rightarrow\big|\overrightarrow{\text{C}}\big|^2=\big|\overrightarrow{\text{A}}-\overrightarrow{\text{B}}\big|^2+4\big|\overrightarrow{\text{A}}\big|.\big|\overrightarrow{\text{B}}\big|\cos120^{\circ}$

Since cosine is negative in the second quadrant, C must be less than $|\text{A}-\text{B}|.$

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