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CBSE Class 12 Math 2020 Outside Delhi Set 1 Solved Paper

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Question : 15
Total: 36
If
→
a
 is a non-zero vector, then (
→
a
⋅
^
i
)
^
i
+(
→
a
⋅
^
j
)
^
j
+(
→
a
⋅
^
k
)
^
k
 equals______
OR
The projection of the vector
^
i
−
^
j
 on the vector
^
i
+
^
j
 is______
Solution:     👈: Video Solution
Let
→
a
=a1
^
𝑖
+a2
^
𝑗
+a3
^
k

Now, taking dot product of
→
a
 with
^
𝑖
, we get
→
a
⋅
^
𝑖
=(a1
^
𝑖
+a2
^
𝑗
+a3
^
k
)⋅
^
𝑖
=a1
^
𝑖
⋅
^
𝑖
+a2
^
𝑗
⋅
^
𝑖
+a3
^
k
⋅
^
𝑖

⇒
→
a
⋅
^
𝑖
=a1
^
𝑖
⋅
^
𝑖
+a2⋅0+a3⋅0(∵
^
𝑗
⋅
^
𝑖
=
^
k
⋅
^
𝑖
=0)
⇒
→
a
⋅
^
𝑖
=a1
Similarly, taking dot product of
→
a
 with
^
𝑗
 and
^
k
, we get
→
a
⋅
^
𝑗
=a2‌ and ‌
→
a
⋅
^
k
=a3
⇒(
→
a
⋅
^
𝑖
)
^
𝑖
+(
→
a
⋅
^
𝑗
)
^
𝑗
+(
→
a
⋅
^
k
)
^
k
=a1
^
𝑖
+a2
^
𝑗
+a3
^
k
=
→
a

If
→
a
 is any non-zero vector, then (
→
a
⋅
^
i
)
^
i
+(
→
a
⋅
^
j
)
^
j
+(
→
a
⋅
^
k
)
^
k
 equals
→
a
.
OR
Let
→
a
=
^
𝑖
−
^
𝑗
 and
→
b
=
^
𝑖
+
^
𝑗
.
Now, projection of vector
→
a
 and
→
b
 is given by,
‌
1
|
→
b
|
(
→
a
⋅
→
b
)=‌
1
√1+1
{1.1+(−1)(1)}=‌
1
√2
(1−1)=0
Hence, the projection of vector
→
a
 on
→
b
 is 0 .
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