Find the vector projection of 5i^ − 4j^ + k^ along the vector 3i^ − 2j^+4k^?
Answers
Answer:
vector projection is
<
−
69
41
,
92
41
,
−
92
41
>
, the scalar projection is
−
23
√
41
41
.
Explanation:
Given
→
a
=
(
3
i
+
2
j
−
6
k
)
and
→
b
=
(
3
i
−
4
j
+
4
k
)
, we can find
p
r
o
j
→
b
→
a
, the vector projection of
→
a
onto
→
b
using the following formula:
p
r
o
j
→
b
→
a
=
⎛
⎜
⎜
⎜
⎝
→
a
⋅
→
b
∣
∣
∣
→
b
∣
∣
∣
⎞
⎟
⎟
⎟
⎠
→
b
∣
∣
∣
→
b
∣
∣
∣
That is, the dot product of the two vectors divided by the magnitude of
→
b
, multiplied by
→
b
divided by its magnitude. The second quantity is a vector quantity, as we divide a vector by a scalar. Note that we divide
→
b
by its magnitude in order to obtain a unit vector (vector with magnitude of
1
). You might notice that the first quantity is scalar, as we know that when we take the dot product of two vectors, the resultant is a scalar.
Therefore, the scalar projection of
a
onto
b
is
c
o
m
p
→
b
→
a
=
a
⋅
b
|
b
|
, also written
∣
∣
p
r
o
j
→
b
→
a
∣
∣
.
We can start by taking the dot product of the two vectors, which can be written as
→
a
=
<
3
,
2
,
−
6
>
and
→
b
=
<
3
,
−
4
,
4
>
.
→
a
⋅
→
b
=
<
3
,
2
,
−
6
>
⋅
<
3
,
−
4
,
4
>
⇒
(
3
⋅
3
)
+
(
2
⋅
−
4
)
+
(
−
6
⋅
4
)
⇒
9
−
8
−
24
=
−
23
Then we can find the magnitude of
→
b
by taking the square root of the sum of the squares of each of the components.
∣
∣
∣
→
b
∣
∣
∣
=
√
(
b
x
)
2
+
(
b
y
)
2
+
(
b
z
)
2
∣
∣
∣
→
b
∣
∣
∣
=
√
(
3
)
2
+
(
−
4
)
2
+
(
4
)
2
⇒
√
9
+
16
+
16
=
√
41
And now we have everything we need to find the vector projection of
→
a
onto
→
b
.
p
r
o
j
→
b
→
a
=
−
23
√
41
⋅
<
3
,
−
4
,
4
>
√
41
⇒
−
23
<
3
,
−
4
,
4
>
41
⇒
−
23
41
<
3
,
−
4
,
4
>
You can distribute the coefficient to each component of the vector and write as:
⇒
<
−
69
41
,
92
41
,
−
92
41
>
The scalar projection of
→
a
onto
→
b
is just the first half of the formula, where
c
o
m
p
→
b
→
a
=
a
⋅
b
|
b
|
. Therefore, the scalar projection is
−
23
√
41
, which does not simplify any further, besides to rationalize the denominator if desired, giving
−
23
√
41
41
.
Explanation:
follow me.
hope it helped you