Find the scalar and vector projections of b onto a. a=(4,7,-4),b=(3,-1,1)

bobbie71G

bobbie71G

Answered question

2021-05-17

Find the scalar and vector projections of b onto a.
a=(4,7,4),b=(3,1,1)

Answer & Explanation

Anonym

Anonym

Skilled2021-05-18Added 108 answers

ab=(4,7,4)(3,1,1) =43+7(1)+(4)1 =1274 =1 
|a|=42+72+(4)2 =16+49+16 =81 =9 
compab=ab|a| 
compab=19 
projab=[compab]a|a| 
projab=[19](4,7,4)9=(481,781,481) 
Answer:  compab=19 projab=(481,781,481)

Jazz Frenia

Jazz Frenia

Skilled2023-06-11Added 106 answers

Result:
(481,781,481)
Solution:
The scalar projection of 𝐛 onto 𝐚 is given by:
Scalar Projection of 𝐛 onto 𝐚=𝐚·𝐛𝐚
The vector projection of 𝐛 onto 𝐚 is calculated as:
Vector Projection of 𝐛 onto 𝐚=(𝐚·𝐛𝐚2)·𝐚
Given 𝐚=(4,7,4) and 𝐛=(3,1,1), let's calculate the scalar and vector projections.
First, we need to find the dot product of 𝐚 and 𝐛:
𝐚·𝐛=(4)(3)+(7)(1)+(4)(1)=1274=1
Next, we calculate the magnitude (norm) of 𝐚:
𝐚=(4)2+(7)2+(4)2=16+49+16=81=9
Now we can substitute the values into the formulas:
The scalar projection of 𝐛 onto 𝐚 is:
Scalar Projection of 𝐛 onto 𝐚=𝐚·𝐛𝐚=19
The vector projection of 𝐛 onto 𝐚 is:
Vector Projection of 𝐛 onto 𝐚=(𝐚·𝐛𝐚2)·𝐚=(181)·(4,7,4)
Simplifying the vector projection, we have:
Vector Projection of 𝐛 onto 𝐚=(481,781,481)
Therefore, the scalar projection of 𝐛 onto 𝐚 is 19, and the vector projection of 𝐛 onto 𝐚 is (481,781,481).
Andre BalkonE

Andre BalkonE

Skilled2023-06-11Added 110 answers

To find the scalar and vector projections of 𝐛 onto 𝐚, we can use the following formulas:
The scalar projection of 𝐛 onto 𝐚 is given by:
Scalar projection of 𝐛 onto 𝐚=𝐚·𝐛|𝐚|
where · represents the dot product and |𝐚| denotes the magnitude of 𝐚.
The vector projection of 𝐛 onto 𝐚 is calculated as:
Vector projection of 𝐛 onto 𝐚=(𝐚·𝐛|𝐚|2)𝐚
Given 𝐚=(4,7,4) and 𝐛=(3,1,1), we can substitute these values into the formulas to find the projections.
First, let's calculate the dot product of 𝐚 and 𝐛:
𝐚·𝐛=(4)(3)+(7)(1)+(4)(1)
Next, we'll compute the magnitude of 𝐚:
|𝐚|=(4)2+(7)2+(4)2
Using these results, we can find the scalar and vector projections of 𝐛 onto 𝐚.

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