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Vector Algebra question

2023 · 15 Apr · Shift 1 · Q27
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Vector Algebra question

2023 · 15 Apr · Shift 1 · Q27

JEE MainMathematicsVector AlgebraMCQ+4 / −1
Let ABCD\mathrm{ABCD}ABCD be a quadrilateral. If E\mathrm{E}E and F\mathrm{F}F are the mid points of the diagonals AC\mathrm{AC}AC and BD\mathrm{BD}BD respectively and (AB→−BC→)+(AD→−DC→)=kFE→(\overrightarrow{A B}-\overrightarrow{B C})+(\overrightarrow{A D}-\overrightarrow{D C})=k \overrightarrow{F E}(AB−BC)+(AD−DC)=kFE, then kkk is equal to :
  1. A
    -2
  2. B
    4
  3. C
    -4
  4. D
    2
View written solutionFree

Correct answer: C

  1. Write the given vector expression in terms of position vectors

Let the position vectors of points A,B,C,DA,B,C,DA,B,C,D be a⃗,b⃗,c⃗,d⃗\vec a,\vec b,\vec c,\vec da,b,c,d respectively.

Then,

AB→=b⃗−a⃗,BC→=c⃗−b⃗,AD→=d⃗−a⃗,DC→=c⃗−d⃗.\overrightarrow{AB}=\vec b-\vec a,\quad \overrightarrow{BC}=\vec c-\vec b,\quad \overrightarrow{AD}=\vec d-\vec a,\quad \overrightarrow{DC}=\vec c-\vec d.AB=b−a,BC=c−b,AD=d−a,DC=c−d.

We need to simplify

(AB→−BC→)+(AD→−DC→).(\overrightarrow{AB}-\overrightarrow{BC})+(\overrightarrow{AD}-\overrightarrow{DC}).(AB−BC)+(AD−DC).

Substitute:

(b⃗−a⃗−(c⃗−b⃗))+(d⃗−a⃗−(c⃗−d⃗)).(\vec b-\vec a-(\vec c-\vec b))+(\vec d-\vec a-(\vec c-\vec d)).(b−a−(c−b))+(d−a−(c−d)). =(b⃗−a⃗−c⃗+b⃗)+(d⃗−a⃗−c⃗+d⃗)=(\vec b-\vec a-\vec c+\vec b)+(\vec d-\vec a-\vec c+\vec d)=(b−a−c+b)+(d−a−c+d) =2b⃗+2d⃗−2a⃗−2c⃗=2\vec b+2\vec d-2\vec a-2\vec c=2b+2d−2a−2c =2((b⃗+d⃗)−(a⃗+c⃗)).=2\big((\vec b+\vec d)-(\vec a+\vec c)\big).=2((b+d)−(a+c)).
  1. Use midpoint formulas

Since EEE is the midpoint of diagonal ACACAC,

e⃗=a⃗+c⃗2.\vec e=\frac{\vec a+\vec c}{2}.e=2a+c​.

Since FFF is the midpoint of diagonal BDBDBD,

f⃗=b⃗+d⃗2.\vec f=\frac{\vec b+\vec d}{2}.f​=2b+d​.

Therefore,

FE→=e⃗−f⃗=a⃗+c⃗2−b⃗+d⃗2=(a⃗+c⃗)−(b⃗+d⃗)2.\overrightarrow{FE}=\vec e-\vec f =\frac{\vec a+\vec c}{2}-\frac{\vec b+\vec d}{2} =\frac{(\vec a+\vec c)-(\vec b+\vec d)}{2}.FE=e−f​=2a+c​−2b+d​=2(a+c)−(b+d)​.

So,

(b⃗+d⃗)−(a⃗+c⃗)=−2FE→.(\vec b+\vec d)-(\vec a+\vec c)=-2\overrightarrow{FE}.(b+d)−(a+c)=−2FE.
  1. Substitute into the simplified expression

From step 1,

(AB→−BC→)+(AD→−DC→)=2((b⃗+d⃗)−(a⃗+c⃗)).(\overrightarrow{AB}-\overrightarrow{BC})+(\overrightarrow{AD}-\overrightarrow{DC}) =2\big((\vec b+\vec d)-(\vec a+\vec c)\big).(AB−BC)+(AD−DC)=2((b+d)−(a+c)).

Using step 2,

=2(−2FE→)=−4FE→.=2(-2\overrightarrow{FE})=-4\overrightarrow{FE}.=2(−2FE)=−4FE.

Hence,

k=−4.k=-4.k=−4.
  1. Check the options
  • A: −2-2−2 ❌
  • B: 444 ❌
  • C: −4-4−4 ✅
  • D: 222 ❌

Therefore, the correct option is C.

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