Linear Algebra: orthogonality in $mathbb{R}^{3}$
$$v_{1} = (1, 1, 1)$$
$$v_{2} = (-1, 1, 0)$$
$$v_{3} = (-1, -1, 2)$$
The vectors $v1$, $v2$, and $v3$ form an orthogonal basis for $mathbb{R}^{3}$.
If $w = (4, -2, 4)$, then what is the coordinate vector, $[w]_{B}$, of $w$ with respect to the basis $B = {v1, v2, v3}$
linear-algebra orthogonality
edited Dec 9 at 4:31
dantopa
6,40932042
asked Dec 8 at 23:25
bradp
1
add a comment |
$$v_{1} = (1, 1, 1)$$
$$v_{2} = (-1, 1, 0)$$
$$v_{3} = (-1, -1, 2)$$
The vectors $v1$, $v2$, and $v3$ form an orthogonal basis for $mathbb{R}^{3}$.
If $w = (4, -2, 4)$, then what is the coordinate vector, $[w]_{B}$, of $w$ with respect to the basis $B = {v1, v2, v3}$
linear-algebra orthogonality
edited Dec 9 at 4:31
dantopa
6,40932042
asked Dec 8 at 23:25
bradp
1
Welcome to math.se. Here are some tips on how to ask a good question.
– André 3000
Dec 9 at 3:43
add a comment |
$$v_{1} = (1, 1, 1)$$
$$v_{2} = (-1, 1, 0)$$
$$v_{3} = (-1, -1, 2)$$
The vectors $v1$, $v2$, and $v3$ form an orthogonal basis for $mathbb{R}^{3}$.
If $w = (4, -2, 4)$, then what is the coordinate vector, $[w]_{B}$, of $w$ with respect to the basis $B = {v1, v2, v3}$
linear-algebra orthogonality
edited Dec 9 at 4:31
dantopa
6,40932042
asked Dec 8 at 23:25
bradp
1
$$v_{1} = (1, 1, 1)$$
$$v_{2} = (-1, 1, 0)$$
$$v_{3} = (-1, -1, 2)$$
The vectors $v1$, $v2$, and $v3$ form an orthogonal basis for $mathbb{R}^{3}$.
If $w = (4, -2, 4)$, then what is the coordinate vector, $[w]_{B}$, of $w$ with respect to the basis $B = {v1, v2, v3}$
linear-algebra orthogonality
linear-algebra orthogonality
edited Dec 9 at 4:31
dantopa
6,40932042
asked Dec 8 at 23:25
bradp
1
edited Dec 9 at 4:31
dantopa
6,40932042
asked Dec 8 at 23:25
bradp
1
edited Dec 9 at 4:31
dantopa
6,40932042
edited Dec 9 at 4:31
dantopa
6,40932042
edited Dec 9 at 4:31
dantopa
6,40932042
6,40932042
asked Dec 8 at 23:25
bradp
1
asked Dec 8 at 23:25
bradp
1
asked Dec 8 at 23:25
bradp
1
1
Welcome to math.se. Here are some tips on how to ask a good question.
– André 3000
Dec 9 at 3:43
add a comment |
Welcome to math.se. Here are some tips on how to ask a good question.
– André 3000
Dec 9 at 3:43
Welcome to math.se. Here are some tips on how to ask a good question.
– André 3000
Dec 9 at 3:43
Welcome to math.se. Here are some tips on how to ask a good question.
– André 3000
Dec 9 at 3:43
add a comment |
1 Answer
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oldest
votes
Suppose
$$(4,-2,4)=w=av_1+bv_2+cv_3implies langle v_1,wrangle = 3a_1$$
Can you now complete the deduction? Can you see why having an orthonormal basis makes things even easier than the above?
answered Dec 8 at 23:37
DonAntonio
176k1491225
what do you mean by this?
– bradp
Dec 8 at 23:43
Where did the 3a1 come from?
– bradp
Dec 8 at 23:46
ok so I think i know where you got 3a1 from. did you just do the dot product of v1 with itself? Then repeated this to find b and c?
– bradp
Dec 8 at 23:55
@bradp Yes, but with $;v_2,v_3;$ the dot product is zero...
– DonAntonio
Dec 9 at 0:51
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Suppose
$$(4,-2,4)=w=av_1+bv_2+cv_3implies langle v_1,wrangle = 3a_1$$
Can you now complete the deduction? Can you see why having an orthonormal basis makes things even easier than the above?
answered Dec 8 at 23:37
DonAntonio
176k1491225
what do you mean by this?
– bradp
Dec 8 at 23:43
Where did the 3a1 come from?
– bradp
Dec 8 at 23:46
ok so I think i know where you got 3a1 from. did you just do the dot product of v1 with itself? Then repeated this to find b and c?
– bradp
Dec 8 at 23:55
@bradp Yes, but with $;v_2,v_3;$ the dot product is zero...
– DonAntonio
Dec 9 at 0:51
add a comment |
Suppose
$$(4,-2,4)=w=av_1+bv_2+cv_3implies langle v_1,wrangle = 3a_1$$
Can you now complete the deduction? Can you see why having an orthonormal basis makes things even easier than the above?
answered Dec 8 at 23:37
DonAntonio
176k1491225
what do you mean by this?
– bradp
Dec 8 at 23:43
Where did the 3a1 come from?
– bradp
Dec 8 at 23:46
ok so I think i know where you got 3a1 from. did you just do the dot product of v1 with itself? Then repeated this to find b and c?
– bradp
Dec 8 at 23:55
@bradp Yes, but with $;v_2,v_3;$ the dot product is zero...
– DonAntonio
Dec 9 at 0:51
add a comment |
Suppose
$$(4,-2,4)=w=av_1+bv_2+cv_3implies langle v_1,wrangle = 3a_1$$
Can you now complete the deduction? Can you see why having an orthonormal basis makes things even easier than the above?
answered Dec 8 at 23:37
DonAntonio
176k1491225
Suppose
$$(4,-2,4)=w=av_1+bv_2+cv_3implies langle v_1,wrangle = 3a_1$$
Can you now complete the deduction? Can you see why having an orthonormal basis makes things even easier than the above?
answered Dec 8 at 23:37
DonAntonio
176k1491225
answered Dec 8 at 23:37
DonAntonio
176k1491225
answered Dec 8 at 23:37
DonAntonio
176k1491225
answered Dec 8 at 23:37
DonAntonio
176k1491225
176k1491225
what do you mean by this?
– bradp
Dec 8 at 23:43
Where did the 3a1 come from?
– bradp
Dec 8 at 23:46
ok so I think i know where you got 3a1 from. did you just do the dot product of v1 with itself? Then repeated this to find b and c?
– bradp
Dec 8 at 23:55
@bradp Yes, but with $;v_2,v_3;$ the dot product is zero...
– DonAntonio
Dec 9 at 0:51
add a comment |
what do you mean by this?
– bradp
Dec 8 at 23:43
Where did the 3a1 come from?
– bradp
Dec 8 at 23:46
ok so I think i know where you got 3a1 from. did you just do the dot product of v1 with itself? Then repeated this to find b and c?
– bradp
Dec 8 at 23:55
@bradp Yes, but with $;v_2,v_3;$ the dot product is zero...
– DonAntonio
Dec 9 at 0:51
what do you mean by this?
– bradp
Dec 8 at 23:43
what do you mean by this?
– bradp
Dec 8 at 23:43
Where did the 3a1 come from?
– bradp
Dec 8 at 23:46
Where did the 3a1 come from?
– bradp
Dec 8 at 23:46
ok so I think i know where you got 3a1 from. did you just do the dot product of v1 with itself? Then repeated this to find b and c?
– bradp
Dec 8 at 23:55
ok so I think i know where you got 3a1 from. did you just do the dot product of v1 with itself? Then repeated this to find b and c?
– bradp
Dec 8 at 23:55
@bradp Yes, but with $;v_2,v_3;$ the dot product is zero...
– DonAntonio
Dec 9 at 0:51
@bradp Yes, but with $;v_2,v_3;$ the dot product is zero...
– DonAntonio
Dec 9 at 0:51
add a comment |
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Welcome to math.se. Here are some tips on how to ask a good question.
– André 3000
Dec 9 at 3:43