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Page 1
Solved Examples on Vector Algebra
JEE Mains
Q1. The length of longer diagonal of the parallelogram constructed on ???? + ????
and ?? - ???? , it is given that |?? | = ?? v?? ,|?? | = ?? and angle between ?? and ?? is
?? ?? , is
(a) 15
(b) v??????
(c) v??????
(d) v??????
Ans: (c) Length of the two diagonals will be ?? 1
= (5?? + 2?? )+ (?? - 3?? ) | and ?? 2
? (5?? +
2?? )- (?? - 3?? ) | ? ?? 1
? 6?? - ?? |,?? 2
? 4?? + 5?? |
Thus, ?? 1
= v6?? |
2
+ | - ?? |
2
+ 2|6?? || - ?? |cos?(?? - ?? /4) =
v36(2v2)
2
+ 9 + 12· 2v2· 3 · (-
1
v2
) = 15.
?? 2
= v4?? |
2
+ |5?? |
2
+ 2|4?? ||5?? |cos?
?? 4
= v16× 8 + 25× 9 + 40× 2v2× 3 ×
1
v2
= v593.
? Length of the longer diagonal = v593
Q.?? The vector ?? , directed along the internal bisector of the angle between the
vectors ?? = ???? - ???? - ???? and ?? = -???? - ?? + ???? with |?? | = ?? v?? , is
(a)
?? ?? (?? - ???? + ???? )
(b)
?? ?? (???? + ???? + ???? )
(c)
?? ?? (?? + ???? + ???? )
(d)
?? ?? (-???? + ???? + ???? )
Ans: (a) Let ?? = 7?? - 4?? - 4?? and ?? = -2?? - ?? + 2??
Now required vector ?? = ?? (
?? |?? |
+
?? |?? |
) = ?? (
7?? -4?? -4?? 9
+
-2?? -?? +2?? 3
) =
?? 9
(?? - 7?? + 2?? )
|?? |
2
=
?? 2
81
× 54 = 150? ?? = ±15 ? ?? = ±
5
3
(?? - 7?? + 2?? )
Q3. The position vectors of the vertices ?? ,?? ,?? of a triangle are ?? - ?? - ???? ,???? + ?? - ????
and -???? + ???? - ???? respectively. The length of the bisector ???? of the angle ??????
where ?? is on the segment ???? , is
(a)
?? ?? v????
(b)
?? ??
(c)
????
??
(d) None of these
Ans: (a)
Page 2
Solved Examples on Vector Algebra
JEE Mains
Q1. The length of longer diagonal of the parallelogram constructed on ???? + ????
and ?? - ???? , it is given that |?? | = ?? v?? ,|?? | = ?? and angle between ?? and ?? is
?? ?? , is
(a) 15
(b) v??????
(c) v??????
(d) v??????
Ans: (c) Length of the two diagonals will be ?? 1
= (5?? + 2?? )+ (?? - 3?? ) | and ?? 2
? (5?? +
2?? )- (?? - 3?? ) | ? ?? 1
? 6?? - ?? |,?? 2
? 4?? + 5?? |
Thus, ?? 1
= v6?? |
2
+ | - ?? |
2
+ 2|6?? || - ?? |cos?(?? - ?? /4) =
v36(2v2)
2
+ 9 + 12· 2v2· 3 · (-
1
v2
) = 15.
?? 2
= v4?? |
2
+ |5?? |
2
+ 2|4?? ||5?? |cos?
?? 4
= v16× 8 + 25× 9 + 40× 2v2× 3 ×
1
v2
= v593.
? Length of the longer diagonal = v593
Q.?? The vector ?? , directed along the internal bisector of the angle between the
vectors ?? = ???? - ???? - ???? and ?? = -???? - ?? + ???? with |?? | = ?? v?? , is
(a)
?? ?? (?? - ???? + ???? )
(b)
?? ?? (???? + ???? + ???? )
(c)
?? ?? (?? + ???? + ???? )
(d)
?? ?? (-???? + ???? + ???? )
Ans: (a) Let ?? = 7?? - 4?? - 4?? and ?? = -2?? - ?? + 2??
Now required vector ?? = ?? (
?? |?? |
+
?? |?? |
) = ?? (
7?? -4?? -4?? 9
+
-2?? -?? +2?? 3
) =
?? 9
(?? - 7?? + 2?? )
|?? |
2
=
?? 2
81
× 54 = 150? ?? = ±15 ? ?? = ±
5
3
(?? - 7?? + 2?? )
Q3. The position vectors of the vertices ?? ,?? ,?? of a triangle are ?? - ?? - ???? ,???? + ?? - ????
and -???? + ???? - ???? respectively. The length of the bisector ???? of the angle ??????
where ?? is on the segment ???? , is
(a)
?? ?? v????
(b)
?? ??
(c)
????
??
(d) None of these
Ans: (a)
?|????
????
| = |(2?? + ?? - 2?? ) - (?? - ?? - 3?? )| = |?? + 2?? + ?? | = v1
2
+ 2
2
+ 1
2
= v6
|????
????
| = |(-5?? + 2?? - 6?? ) - (?? - ?? - 3?? )| = | - 6??ˆ + 3??ˆ - 3?? ˆ
| = v(-6)
2
+ 3
2
+ (-3)
2
? = v54 = 3v6.
???? : ???? = ???? : ???? =
v6
3v6
=
1
3
.
?? ? Position vector of ?? =
1. (-5?? + 2?? - 6?? ) + 3(2?? + ?? - 2?? )
1 + 3
=
1
4
(?? + 5?? - 12?? )
?? ????
????
= position vector of ?? - Position vector of ?? =
1
4
(?? + 5?? - 12?? ) - (?? - ?? - 3?? ) =
1
4
(-3?? + 9?? ) =
3
4
(-?? + 3?? )
?|????
????
| =
3
4
v(-1)
2
+ 3
2
=
3
4
v10.
Q4. The median ???? of the triangle ?????? is bisected at ?? ,???? meets ???? in ?? . Then
???? :???? =
(a) ?? /??
(b) ?? /??
(c) ?? /??
(d) ?? /??
Ans: (b) Let position vector of ?? with respect to ?? is a and that of ?? w.r.t. ?? ic a
Position vector of ?? w.r.t. ?? =
?? +?? 2
=
?? 2
Position vector of ?? =
?? +
?? 2
2
=
?? 2
+
?? 4
Let ???? :???? = ?? :1 and ???? :???? = ?? :1
(0)
Position vector of ?? =
?? ?? +?? 1+??
Now, position vector of ?? =
?? (
?? ?? +?? 1+?? )+1.0
?? +1
Page 3
Solved Examples on Vector Algebra
JEE Mains
Q1. The length of longer diagonal of the parallelogram constructed on ???? + ????
and ?? - ???? , it is given that |?? | = ?? v?? ,|?? | = ?? and angle between ?? and ?? is
?? ?? , is
(a) 15
(b) v??????
(c) v??????
(d) v??????
Ans: (c) Length of the two diagonals will be ?? 1
= (5?? + 2?? )+ (?? - 3?? ) | and ?? 2
? (5?? +
2?? )- (?? - 3?? ) | ? ?? 1
? 6?? - ?? |,?? 2
? 4?? + 5?? |
Thus, ?? 1
= v6?? |
2
+ | - ?? |
2
+ 2|6?? || - ?? |cos?(?? - ?? /4) =
v36(2v2)
2
+ 9 + 12· 2v2· 3 · (-
1
v2
) = 15.
?? 2
= v4?? |
2
+ |5?? |
2
+ 2|4?? ||5?? |cos?
?? 4
= v16× 8 + 25× 9 + 40× 2v2× 3 ×
1
v2
= v593.
? Length of the longer diagonal = v593
Q.?? The vector ?? , directed along the internal bisector of the angle between the
vectors ?? = ???? - ???? - ???? and ?? = -???? - ?? + ???? with |?? | = ?? v?? , is
(a)
?? ?? (?? - ???? + ???? )
(b)
?? ?? (???? + ???? + ???? )
(c)
?? ?? (?? + ???? + ???? )
(d)
?? ?? (-???? + ???? + ???? )
Ans: (a) Let ?? = 7?? - 4?? - 4?? and ?? = -2?? - ?? + 2??
Now required vector ?? = ?? (
?? |?? |
+
?? |?? |
) = ?? (
7?? -4?? -4?? 9
+
-2?? -?? +2?? 3
) =
?? 9
(?? - 7?? + 2?? )
|?? |
2
=
?? 2
81
× 54 = 150? ?? = ±15 ? ?? = ±
5
3
(?? - 7?? + 2?? )
Q3. The position vectors of the vertices ?? ,?? ,?? of a triangle are ?? - ?? - ???? ,???? + ?? - ????
and -???? + ???? - ???? respectively. The length of the bisector ???? of the angle ??????
where ?? is on the segment ???? , is
(a)
?? ?? v????
(b)
?? ??
(c)
????
??
(d) None of these
Ans: (a)
?|????
????
| = |(2?? + ?? - 2?? ) - (?? - ?? - 3?? )| = |?? + 2?? + ?? | = v1
2
+ 2
2
+ 1
2
= v6
|????
????
| = |(-5?? + 2?? - 6?? ) - (?? - ?? - 3?? )| = | - 6??ˆ + 3??ˆ - 3?? ˆ
| = v(-6)
2
+ 3
2
+ (-3)
2
? = v54 = 3v6.
???? : ???? = ???? : ???? =
v6
3v6
=
1
3
.
?? ? Position vector of ?? =
1. (-5?? + 2?? - 6?? ) + 3(2?? + ?? - 2?? )
1 + 3
=
1
4
(?? + 5?? - 12?? )
?? ????
????
= position vector of ?? - Position vector of ?? =
1
4
(?? + 5?? - 12?? ) - (?? - ?? - 3?? ) =
1
4
(-3?? + 9?? ) =
3
4
(-?? + 3?? )
?|????
????
| =
3
4
v(-1)
2
+ 3
2
=
3
4
v10.
Q4. The median ???? of the triangle ?????? is bisected at ?? ,???? meets ???? in ?? . Then
???? :???? =
(a) ?? /??
(b) ?? /??
(c) ?? /??
(d) ?? /??
Ans: (b) Let position vector of ?? with respect to ?? is a and that of ?? w.r.t. ?? ic a
Position vector of ?? w.r.t. ?? =
?? +?? 2
=
?? 2
Position vector of ?? =
?? +
?? 2
2
=
?? 2
+
?? 4
Let ???? :???? = ?? :1 and ???? :???? = ?? :1
(0)
Position vector of ?? =
?? ?? +?? 1+??
Now, position vector of ?? =
?? (
?? ?? +?? 1+?? )+1.0
?? +1
?? 2
+
?? 4
=
?? (1+ ?? )(1+ ?? )
?? +
????
(1+ ?? )(1+ ?? )
?? ? ?
1
2
=
?? (1+ ?? )(1+ ?? )
and
1
4
=
????
(1+ ?? )(1+ ?? )
? ?? =
1
2
,?
????
????
=
????
???? + ????
=
?? 1+ ?? =
1
2
3
2
=
1
3
.
Q5. The points with position vectors ?????? + ???? ,?????? - ???? ,?? ?? - ?????? are collinear, if ?? =
(a) -40
(b) 40
(c) 20
(d) None of these
Ans: (a) As the three points are collinear, ?? (60?? + 3?? )+ ?? (40?? - 8?? )+ ?? (?? ?? - 52?? ) = 0 such
that ?? ,?? ,?? are not all zero and ?? + ?? + ?? = 0.
?? (60?? + 40?? + ???? )?? + (3?? - 8?? - 52?? )?? = 0 and ?? + ?? + ?? = 0
?? 60?? + 40?? + ???? = 0,3?? - 8?? - 52?? = 0 and ?? + ?? + ?? = 0
For non-trivial solution, |
60 40 ?? 3 -8 -52
1 1 1
| = 0 ? ?? = -40
Trick : If ?? ,?? ,?? are given points, then ????
?????
= ?? (????
?????
) ? -20?? - 11?? = ?? [(?? - 40)?? - 44?? ]
On comparing, -11= -44?? ? ?? =
1
4
and -20 =
1
4
(?? - 40) ? ?? = -40.
Q6. If the position vectors of ?? ,?? ,?? ,?? are ???? + ?? ,?? - ???? ,???? + ???? and ?? + ?? ?? respectively
and ????
??????
? ????
??????
, then ?? will be
(a) -8
(b) -6
(c) 8
(d) 6
Ans: (b)
????
?????
= (?? - 3?? )- (2?? + ?? ) = -?? - 4?? ;?????
?????
= (?? + ?? ?? )- (3?? + 2?? ) = -2?? + (?? - 2)?? ;????
?????
? ????
?????
? ????
?????
= ?? ????
?????
?-?? - 4?? = ?? {-2?? + (?? - 2)?? } ? -1 = -2?? ,-4 = (?? - 2)?? ? ?? =
1
2
,?? = -6.
Q7. Let ?? ,?? and ?? be three non-zero vectors such that no two of these are
collinear. If the vector ?? + ???? is collinear with ?? and ?? + ???? is collinear with ?? ( ??
being some non-zero scalar) then ?? + ???? + ???? equals
(a) 0
(b) ?? ??
(c) ?? ??
(d) ?? ??
Ans: (a) As ?? + 2?? and ?? are collinear ?? + 2?? = ?? ??
Again ?? + 3?? is collinear with ??
? ?? + 3?? = ?? ??
Now, ?? + 2?? + 6?? = (?? + 2?? )+ 6?? = ?? ?? + 6?? = (?? + 6)??
Also, ?? + 2?? + 6?? = ?? + 2(?? + 3?? ) = ?? + 2?? ?? = (2?? + 1)?? ?
Page 4
Solved Examples on Vector Algebra
JEE Mains
Q1. The length of longer diagonal of the parallelogram constructed on ???? + ????
and ?? - ???? , it is given that |?? | = ?? v?? ,|?? | = ?? and angle between ?? and ?? is
?? ?? , is
(a) 15
(b) v??????
(c) v??????
(d) v??????
Ans: (c) Length of the two diagonals will be ?? 1
= (5?? + 2?? )+ (?? - 3?? ) | and ?? 2
? (5?? +
2?? )- (?? - 3?? ) | ? ?? 1
? 6?? - ?? |,?? 2
? 4?? + 5?? |
Thus, ?? 1
= v6?? |
2
+ | - ?? |
2
+ 2|6?? || - ?? |cos?(?? - ?? /4) =
v36(2v2)
2
+ 9 + 12· 2v2· 3 · (-
1
v2
) = 15.
?? 2
= v4?? |
2
+ |5?? |
2
+ 2|4?? ||5?? |cos?
?? 4
= v16× 8 + 25× 9 + 40× 2v2× 3 ×
1
v2
= v593.
? Length of the longer diagonal = v593
Q.?? The vector ?? , directed along the internal bisector of the angle between the
vectors ?? = ???? - ???? - ???? and ?? = -???? - ?? + ???? with |?? | = ?? v?? , is
(a)
?? ?? (?? - ???? + ???? )
(b)
?? ?? (???? + ???? + ???? )
(c)
?? ?? (?? + ???? + ???? )
(d)
?? ?? (-???? + ???? + ???? )
Ans: (a) Let ?? = 7?? - 4?? - 4?? and ?? = -2?? - ?? + 2??
Now required vector ?? = ?? (
?? |?? |
+
?? |?? |
) = ?? (
7?? -4?? -4?? 9
+
-2?? -?? +2?? 3
) =
?? 9
(?? - 7?? + 2?? )
|?? |
2
=
?? 2
81
× 54 = 150? ?? = ±15 ? ?? = ±
5
3
(?? - 7?? + 2?? )
Q3. The position vectors of the vertices ?? ,?? ,?? of a triangle are ?? - ?? - ???? ,???? + ?? - ????
and -???? + ???? - ???? respectively. The length of the bisector ???? of the angle ??????
where ?? is on the segment ???? , is
(a)
?? ?? v????
(b)
?? ??
(c)
????
??
(d) None of these
Ans: (a)
?|????
????
| = |(2?? + ?? - 2?? ) - (?? - ?? - 3?? )| = |?? + 2?? + ?? | = v1
2
+ 2
2
+ 1
2
= v6
|????
????
| = |(-5?? + 2?? - 6?? ) - (?? - ?? - 3?? )| = | - 6??ˆ + 3??ˆ - 3?? ˆ
| = v(-6)
2
+ 3
2
+ (-3)
2
? = v54 = 3v6.
???? : ???? = ???? : ???? =
v6
3v6
=
1
3
.
?? ? Position vector of ?? =
1. (-5?? + 2?? - 6?? ) + 3(2?? + ?? - 2?? )
1 + 3
=
1
4
(?? + 5?? - 12?? )
?? ????
????
= position vector of ?? - Position vector of ?? =
1
4
(?? + 5?? - 12?? ) - (?? - ?? - 3?? ) =
1
4
(-3?? + 9?? ) =
3
4
(-?? + 3?? )
?|????
????
| =
3
4
v(-1)
2
+ 3
2
=
3
4
v10.
Q4. The median ???? of the triangle ?????? is bisected at ?? ,???? meets ???? in ?? . Then
???? :???? =
(a) ?? /??
(b) ?? /??
(c) ?? /??
(d) ?? /??
Ans: (b) Let position vector of ?? with respect to ?? is a and that of ?? w.r.t. ?? ic a
Position vector of ?? w.r.t. ?? =
?? +?? 2
=
?? 2
Position vector of ?? =
?? +
?? 2
2
=
?? 2
+
?? 4
Let ???? :???? = ?? :1 and ???? :???? = ?? :1
(0)
Position vector of ?? =
?? ?? +?? 1+??
Now, position vector of ?? =
?? (
?? ?? +?? 1+?? )+1.0
?? +1
?? 2
+
?? 4
=
?? (1+ ?? )(1+ ?? )
?? +
????
(1+ ?? )(1+ ?? )
?? ? ?
1
2
=
?? (1+ ?? )(1+ ?? )
and
1
4
=
????
(1+ ?? )(1+ ?? )
? ?? =
1
2
,?
????
????
=
????
???? + ????
=
?? 1+ ?? =
1
2
3
2
=
1
3
.
Q5. The points with position vectors ?????? + ???? ,?????? - ???? ,?? ?? - ?????? are collinear, if ?? =
(a) -40
(b) 40
(c) 20
(d) None of these
Ans: (a) As the three points are collinear, ?? (60?? + 3?? )+ ?? (40?? - 8?? )+ ?? (?? ?? - 52?? ) = 0 such
that ?? ,?? ,?? are not all zero and ?? + ?? + ?? = 0.
?? (60?? + 40?? + ???? )?? + (3?? - 8?? - 52?? )?? = 0 and ?? + ?? + ?? = 0
?? 60?? + 40?? + ???? = 0,3?? - 8?? - 52?? = 0 and ?? + ?? + ?? = 0
For non-trivial solution, |
60 40 ?? 3 -8 -52
1 1 1
| = 0 ? ?? = -40
Trick : If ?? ,?? ,?? are given points, then ????
?????
= ?? (????
?????
) ? -20?? - 11?? = ?? [(?? - 40)?? - 44?? ]
On comparing, -11= -44?? ? ?? =
1
4
and -20 =
1
4
(?? - 40) ? ?? = -40.
Q6. If the position vectors of ?? ,?? ,?? ,?? are ???? + ?? ,?? - ???? ,???? + ???? and ?? + ?? ?? respectively
and ????
??????
? ????
??????
, then ?? will be
(a) -8
(b) -6
(c) 8
(d) 6
Ans: (b)
????
?????
= (?? - 3?? )- (2?? + ?? ) = -?? - 4?? ;?????
?????
= (?? + ?? ?? )- (3?? + 2?? ) = -2?? + (?? - 2)?? ;????
?????
? ????
?????
? ????
?????
= ?? ????
?????
?-?? - 4?? = ?? {-2?? + (?? - 2)?? } ? -1 = -2?? ,-4 = (?? - 2)?? ? ?? =
1
2
,?? = -6.
Q7. Let ?? ,?? and ?? be three non-zero vectors such that no two of these are
collinear. If the vector ?? + ???? is collinear with ?? and ?? + ???? is collinear with ?? ( ??
being some non-zero scalar) then ?? + ???? + ???? equals
(a) 0
(b) ?? ??
(c) ?? ??
(d) ?? ??
Ans: (a) As ?? + 2?? and ?? are collinear ?? + 2?? = ?? ??
Again ?? + 3?? is collinear with ??
? ?? + 3?? = ?? ??
Now, ?? + 2?? + 6?? = (?? + 2?? )+ 6?? = ?? ?? + 6?? = (?? + 6)??
Also, ?? + 2?? + 6?? = ?? + 2(?? + 3?? ) = ?? + 2?? ?? = (2?? + 1)?? ?
Q8. The value of ?? for which the four points ???? + ???? - ?? ,?? + ???? + ???? ,???? + ???? - ???? ,?? -
?? ?? + ???? are coplanar
(a) 8
(b) 0
(c) -2
(d) 6
Ans: (c) The given four points are coplanar
? ?? (2?? + 3?? - ?? )+ ?? (?? + 2?? + 3?? )+ ?? (3?? + 4?? - 2?? )+ ?? (?? - ?? ?? + 6?? ) = ?? and ?? + ?? + ?? + ?? = 0,
where ?? ,?? ,?? ,?? are not all zero.
?? ?(2?? + ?? + 3?? + ?? )?? + (3?? + 2?? + 4?? - ???? )?? + (-?? + 3?? - 2?? + 6?? )?? = 0 and ?? + ?? + ?? + ?? = 0
?? ?2?? + ?? + 3?? + ?? = 0,3?? + 2?? + 4?? - ???? = 0,-?? + 3?? - 2?? + 6?? = 0 and ?? + ?? + ?? + ?? = 0
For non-trivial solution, |
2 1 3 1
3 2 4 -?? -1 3 -2 6
1 1 1 1
| = 0 ? |
2 1 3 1
0 0 0 -(?? + 2)
-1 3 -2 6
1 1 1 1
| = 0, Operating
[?? 2
? ?? 2
- ?? 1
- ?? 4
?
? -(?? + 2)|
2 1 3
-1 3 -2
1 1 1
| = 0 ? ?? = -2.
Q9. If ?? = ?? + ?? + ?? ,??? = ???? + ???? + ???? and ?? = ?? + ?? ?? + ?? ?? are linearly dependent
vectors and |?? | = v?? , then
(a) ?? = ?? ,?? = -??
(b) ?? = ?? ,?? = ±??
(c) ?? = -?? ,?? = ±??
(d) ?? = ±?? ,?? = ??
Ans: (d) The given vectors are linearly dependent hence, there exist scalars ?? ,?? ,?? not all
zero, such that
?? ?? + ?? ?? + ?? ?? = ?? i.e., ?? (?? + ?? + ?? )+ ?? (4?? + 3?? + 4?? )+ ?? (?? + ?? ?? + ?? ?? )= ?? ,
i.e., (?? + 4?? + ?? )?? + (?? + 3?? + ???? )?? + (?? + 4?? + ???? )?? = ?? ?? ?? + 4?? + ?? = 0,?? + 3?? + ???? = 0,?? + 4?? + ???? = 0
For non-trivial solution, |
1 4 1
1 3 ?? 1 4 ?? | = 0 ? ?? = 1
|?? |
2
= 3 ? 1+ ?? 2
+ ?? 2
= 3 ? ?? 2
= 2- ?? 2
= 2- 1 = 1;? ? ??? = ±1
Page 5
Solved Examples on Vector Algebra
JEE Mains
Q1. The length of longer diagonal of the parallelogram constructed on ???? + ????
and ?? - ???? , it is given that |?? | = ?? v?? ,|?? | = ?? and angle between ?? and ?? is
?? ?? , is
(a) 15
(b) v??????
(c) v??????
(d) v??????
Ans: (c) Length of the two diagonals will be ?? 1
= (5?? + 2?? )+ (?? - 3?? ) | and ?? 2
? (5?? +
2?? )- (?? - 3?? ) | ? ?? 1
? 6?? - ?? |,?? 2
? 4?? + 5?? |
Thus, ?? 1
= v6?? |
2
+ | - ?? |
2
+ 2|6?? || - ?? |cos?(?? - ?? /4) =
v36(2v2)
2
+ 9 + 12· 2v2· 3 · (-
1
v2
) = 15.
?? 2
= v4?? |
2
+ |5?? |
2
+ 2|4?? ||5?? |cos?
?? 4
= v16× 8 + 25× 9 + 40× 2v2× 3 ×
1
v2
= v593.
? Length of the longer diagonal = v593
Q.?? The vector ?? , directed along the internal bisector of the angle between the
vectors ?? = ???? - ???? - ???? and ?? = -???? - ?? + ???? with |?? | = ?? v?? , is
(a)
?? ?? (?? - ???? + ???? )
(b)
?? ?? (???? + ???? + ???? )
(c)
?? ?? (?? + ???? + ???? )
(d)
?? ?? (-???? + ???? + ???? )
Ans: (a) Let ?? = 7?? - 4?? - 4?? and ?? = -2?? - ?? + 2??
Now required vector ?? = ?? (
?? |?? |
+
?? |?? |
) = ?? (
7?? -4?? -4?? 9
+
-2?? -?? +2?? 3
) =
?? 9
(?? - 7?? + 2?? )
|?? |
2
=
?? 2
81
× 54 = 150? ?? = ±15 ? ?? = ±
5
3
(?? - 7?? + 2?? )
Q3. The position vectors of the vertices ?? ,?? ,?? of a triangle are ?? - ?? - ???? ,???? + ?? - ????
and -???? + ???? - ???? respectively. The length of the bisector ???? of the angle ??????
where ?? is on the segment ???? , is
(a)
?? ?? v????
(b)
?? ??
(c)
????
??
(d) None of these
Ans: (a)
?|????
????
| = |(2?? + ?? - 2?? ) - (?? - ?? - 3?? )| = |?? + 2?? + ?? | = v1
2
+ 2
2
+ 1
2
= v6
|????
????
| = |(-5?? + 2?? - 6?? ) - (?? - ?? - 3?? )| = | - 6??ˆ + 3??ˆ - 3?? ˆ
| = v(-6)
2
+ 3
2
+ (-3)
2
? = v54 = 3v6.
???? : ???? = ???? : ???? =
v6
3v6
=
1
3
.
?? ? Position vector of ?? =
1. (-5?? + 2?? - 6?? ) + 3(2?? + ?? - 2?? )
1 + 3
=
1
4
(?? + 5?? - 12?? )
?? ????
????
= position vector of ?? - Position vector of ?? =
1
4
(?? + 5?? - 12?? ) - (?? - ?? - 3?? ) =
1
4
(-3?? + 9?? ) =
3
4
(-?? + 3?? )
?|????
????
| =
3
4
v(-1)
2
+ 3
2
=
3
4
v10.
Q4. The median ???? of the triangle ?????? is bisected at ?? ,???? meets ???? in ?? . Then
???? :???? =
(a) ?? /??
(b) ?? /??
(c) ?? /??
(d) ?? /??
Ans: (b) Let position vector of ?? with respect to ?? is a and that of ?? w.r.t. ?? ic a
Position vector of ?? w.r.t. ?? =
?? +?? 2
=
?? 2
Position vector of ?? =
?? +
?? 2
2
=
?? 2
+
?? 4
Let ???? :???? = ?? :1 and ???? :???? = ?? :1
(0)
Position vector of ?? =
?? ?? +?? 1+??
Now, position vector of ?? =
?? (
?? ?? +?? 1+?? )+1.0
?? +1
?? 2
+
?? 4
=
?? (1+ ?? )(1+ ?? )
?? +
????
(1+ ?? )(1+ ?? )
?? ? ?
1
2
=
?? (1+ ?? )(1+ ?? )
and
1
4
=
????
(1+ ?? )(1+ ?? )
? ?? =
1
2
,?
????
????
=
????
???? + ????
=
?? 1+ ?? =
1
2
3
2
=
1
3
.
Q5. The points with position vectors ?????? + ???? ,?????? - ???? ,?? ?? - ?????? are collinear, if ?? =
(a) -40
(b) 40
(c) 20
(d) None of these
Ans: (a) As the three points are collinear, ?? (60?? + 3?? )+ ?? (40?? - 8?? )+ ?? (?? ?? - 52?? ) = 0 such
that ?? ,?? ,?? are not all zero and ?? + ?? + ?? = 0.
?? (60?? + 40?? + ???? )?? + (3?? - 8?? - 52?? )?? = 0 and ?? + ?? + ?? = 0
?? 60?? + 40?? + ???? = 0,3?? - 8?? - 52?? = 0 and ?? + ?? + ?? = 0
For non-trivial solution, |
60 40 ?? 3 -8 -52
1 1 1
| = 0 ? ?? = -40
Trick : If ?? ,?? ,?? are given points, then ????
?????
= ?? (????
?????
) ? -20?? - 11?? = ?? [(?? - 40)?? - 44?? ]
On comparing, -11= -44?? ? ?? =
1
4
and -20 =
1
4
(?? - 40) ? ?? = -40.
Q6. If the position vectors of ?? ,?? ,?? ,?? are ???? + ?? ,?? - ???? ,???? + ???? and ?? + ?? ?? respectively
and ????
??????
? ????
??????
, then ?? will be
(a) -8
(b) -6
(c) 8
(d) 6
Ans: (b)
????
?????
= (?? - 3?? )- (2?? + ?? ) = -?? - 4?? ;?????
?????
= (?? + ?? ?? )- (3?? + 2?? ) = -2?? + (?? - 2)?? ;????
?????
? ????
?????
? ????
?????
= ?? ????
?????
?-?? - 4?? = ?? {-2?? + (?? - 2)?? } ? -1 = -2?? ,-4 = (?? - 2)?? ? ?? =
1
2
,?? = -6.
Q7. Let ?? ,?? and ?? be three non-zero vectors such that no two of these are
collinear. If the vector ?? + ???? is collinear with ?? and ?? + ???? is collinear with ?? ( ??
being some non-zero scalar) then ?? + ???? + ???? equals
(a) 0
(b) ?? ??
(c) ?? ??
(d) ?? ??
Ans: (a) As ?? + 2?? and ?? are collinear ?? + 2?? = ?? ??
Again ?? + 3?? is collinear with ??
? ?? + 3?? = ?? ??
Now, ?? + 2?? + 6?? = (?? + 2?? )+ 6?? = ?? ?? + 6?? = (?? + 6)??
Also, ?? + 2?? + 6?? = ?? + 2(?? + 3?? ) = ?? + 2?? ?? = (2?? + 1)?? ?
Q8. The value of ?? for which the four points ???? + ???? - ?? ,?? + ???? + ???? ,???? + ???? - ???? ,?? -
?? ?? + ???? are coplanar
(a) 8
(b) 0
(c) -2
(d) 6
Ans: (c) The given four points are coplanar
? ?? (2?? + 3?? - ?? )+ ?? (?? + 2?? + 3?? )+ ?? (3?? + 4?? - 2?? )+ ?? (?? - ?? ?? + 6?? ) = ?? and ?? + ?? + ?? + ?? = 0,
where ?? ,?? ,?? ,?? are not all zero.
?? ?(2?? + ?? + 3?? + ?? )?? + (3?? + 2?? + 4?? - ???? )?? + (-?? + 3?? - 2?? + 6?? )?? = 0 and ?? + ?? + ?? + ?? = 0
?? ?2?? + ?? + 3?? + ?? = 0,3?? + 2?? + 4?? - ???? = 0,-?? + 3?? - 2?? + 6?? = 0 and ?? + ?? + ?? + ?? = 0
For non-trivial solution, |
2 1 3 1
3 2 4 -?? -1 3 -2 6
1 1 1 1
| = 0 ? |
2 1 3 1
0 0 0 -(?? + 2)
-1 3 -2 6
1 1 1 1
| = 0, Operating
[?? 2
? ?? 2
- ?? 1
- ?? 4
?
? -(?? + 2)|
2 1 3
-1 3 -2
1 1 1
| = 0 ? ?? = -2.
Q9. If ?? = ?? + ?? + ?? ,??? = ???? + ???? + ???? and ?? = ?? + ?? ?? + ?? ?? are linearly dependent
vectors and |?? | = v?? , then
(a) ?? = ?? ,?? = -??
(b) ?? = ?? ,?? = ±??
(c) ?? = -?? ,?? = ±??
(d) ?? = ±?? ,?? = ??
Ans: (d) The given vectors are linearly dependent hence, there exist scalars ?? ,?? ,?? not all
zero, such that
?? ?? + ?? ?? + ?? ?? = ?? i.e., ?? (?? + ?? + ?? )+ ?? (4?? + 3?? + 4?? )+ ?? (?? + ?? ?? + ?? ?? )= ?? ,
i.e., (?? + 4?? + ?? )?? + (?? + 3?? + ???? )?? + (?? + 4?? + ???? )?? = ?? ?? ?? + 4?? + ?? = 0,?? + 3?? + ???? = 0,?? + 4?? + ???? = 0
For non-trivial solution, |
1 4 1
1 3 ?? 1 4 ?? | = 0 ? ?? = 1
|?? |
2
= 3 ? 1+ ?? 2
+ ?? 2
= 3 ? ?? 2
= 2- ?? 2
= 2- 1 = 1;? ? ??? = ±1
Trick: |?? | = v1 + ?? 2
+ ?? 2
= v3 ? ?? 2
+ ?? 2
= 2
? ??? ,?? ,?? are linearly dependent, hence |
1 1 1
4 3 4
1 ?? ?? | = 0 ? ?? = 1.
? ??? 2
= 1 ? ?? = ±1.
Q10. If |?? | = ?? ,?? |= ?? then a value of ?? for which ?? + ?? ?? is perpendicular to ?? - ?? ??
is
(a) ?? /????
(b) ?? /??
(c) ?? /??
(d) ?? /??
Ans: (b) ?? + ?? ?? is perpendicular to ?? - ?? ??
? ?(?? + ?? ?? )· (?? - ?? ?? ) = 0 ? |?? |
2
- ?? (?? · ?? )+ ?? (?? · ?? )- ?? 2
|?? |
2
= 0 ? |?? |
2
- ?? 2
|?? |
2
= 0 ? ?? = ±
|?? |
|?? |
= ±
3
4
Q11. The vectors ?? = ?? ?? ?? ?? + ?? ?? ?? + ?? and ?? = ???? - ???? + ?? ?? make an obtuse angle
whereas the angle between ?? and ?? is acute and less than ?? /?? , then domain of ??
is
(a) ?? < ?? <
?? ??
(b) ?? > v??????
(c) -
?? ?? < ?? < ??
(d) Null set
Ans: (d) As angle between ?? and ?? is obtuse, ?? .?? < 0
? (2?? 2
?? + 4?? ?? + ?? )· (7?? - 2?? + ?? ?? ) < 0 ? 14?? 2
- 8?? + ?? < 0 ? ?? (2?? - 1) < 0 ? 0 < ?? <
1
2
Angle between ?? and ?? is acute and less than
?? 6
.
?? · ?? = |?? | · |?? |cos? ?? ? ?? = v53 + ?? 2
· 1 · cos? ?? ? cos? ?? =
?? v53 + ?? 2
?? <
?? 6
? cos? ?? > cos?
?? 6
? cos? ?? >
v3
2
?
?? v53 + ?? 2
>
v3
2
? 4?? 2
- 3(53 + ?? 2
) > 0 ? ?? 2
> 159 ? ?? < -v159
or ?? > v159
From (i) and (ii), ?? = ?? .? ? Domain of ?? is null set.
Q12. If three non-zero vectors are ?? = ?? ?? ?? + ?? ?? ?? + ?? ?? ?? ,?? = ?? ?? ?? + ?? ?? ?? + ?? ?? ?? and ?? =
?? ?? ?? + ?? ?? ?? + ?? ?? ?? . If ?? is the unit vector perpendicular to the vectors ?? and ?? and the
angle between ?? and ?? is
?? ?? , then |
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? |
?? is equal to
(a) 0
(b)
?? (?? ?? ?? ?? )(?? ?? ?? ?? )(?? ?? ?? ?? )
??
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