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Byju's Answer
Standard XII
Mathematics
Linear Combination of Vectors
If a →=5 i ∧-...
Question
If
a
→
=
5
i
^
-
j
^
-
3
k
^
and
b
→
=
i
^
+
3
j
^
-
5
k
^
,
then show that the vectors
a
→
+
b
→
and
a
→
-
b
→
are orthogonal.
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Solution
Given that
a
→
=
5
i
^
-
j
^
-
3
k
^
;
b
→
=
i
^
+
3
j
^
-
5
k
^
∴
a
→
+
b
→
=
5
i
^
-
j
^
-
3
k
^
+
i
^
+
3
j
^
-
5
k
^
=
6
i
^
+
2
j
^
-
8
k
^
And
a
→
-
b
→
=
5
i
^
-
j
^
-
3
k
^
-
i
^
+
3
j
^
-
5
k
^
=
4
i
^
-
4
j
^
+
2
k
^
Now,
a
→
+
b
→
.
a
→
-
b
→
=
6
i
^
+
2
j
^
-
8
k
^
.
4
i
^
-
4
j
^
+
2
k
^
=
24
-
8
-
16
=
0
So,
a
→
+
b
→
is orthogonal to
a
→
-
b
→
.
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0
Similar questions
Q.
State the whether given statement is true or false
→
a
=
5
ˆ
i
−
ˆ
j
−
3
ˆ
k
and
→
b
=
ˆ
i
+
3
ˆ
j
−
5
ˆ
k
then show that the vectors
→
a
+
→
b
and
→
a
−
→
b
are perpendicular
Q.
Show the each of the following triads of vectors are coplanar:
(i)
a
→
=
i
^
+
2
j
^
-
k
^
,
b
→
=
3
i
^
+
2
j
^
+
7
k
^
,
c
→
=
5
i
^
+
6
j
^
+
5
k
^
(ii)
a
→
=
-
4
i
^
-
6
j
^
-
2
k
^
,
b
→
=
-
i
^
+
4
j
^
+
3
k
^
,
c
→
=
-
8
i
^
-
j
^
+
3
k
^
(iii)
a
^
=
i
^
-
2
j
^
+
3
k
^
,
b
^
=
-
2
i
^
+
3
j
^
-
4
k
^
,
c
^
=
i
^
-
3
j
^
+
5
k
^
Q.
If
a
→
=
i
^
-
2
j
^
+
3
k
^
,
and
b
→
=
2
i
^
+
3
j
^
-
5
k
^
,
then find
a
→
×
b
→
.
Verify that
a
→
and
a
→
×
b
→
are perpendicular to each other.
Q.
If
p
→
=
5
i
^
+
λ
j
^
-
3
k
^
and
q
→
=
i
^
+
3
j
^
-
5
k
^
,
then find the value of λ, so that
p
→
+
q
→
and
p
→
-
q
→
are perpendicular vectors.
Q.
Find the value of λ so that the following vectors are coplanar:
(i)
a
→
=
i
^
-
j
^
+
k
^
,
b
→
=
2
i
^
+
j
^
-
k
^
,
c
→
=
λ
i
^
-
j
^
+
λ
k
^
(ii)
a
→
=
2
i
^
-
j
^
+
k
^
,
b
→
=
i
^
+
2
j
^
-
3
k
^
,
c
→
=
λ
i
^
+
λ
j
^
+
5
k
^
(iii)
a
→
=
i
^
+
2
j
^
-
3
k
^
,
b
→
=
3
i
^
+
λ
j
^
+
k
^
,
c
→
=
i
^
+
2
j
^
+
2
k
^
(iv)
a
→
=
i
^
+
3
j
^
,
b
→
=
5
k
^
,
c
→
=
λ
i
^
-
j
^
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