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junior college 2 | H2 Maths
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Hi! can someone pls help me with this vector question? i’m not sure which formulas i should apply to determine if a b c are coplanar.. Thanks so much:)
Date Posted: 4 years ago
Views: 342
if vectors a, b and c are coplanar, then any one of the 3 vectors, can be expressed as sum of scalar multiples of the other 2 vectors,
eg, c = ma + nb, where m & n are real numbers.
the first part of the question gives us the possible way to solve the second part....
1) we assume that the 3 vectors are coplanar, then c = ma + nb.
2) by comparing the i & j components in the equation, solve for m & n.
3) using the values of m & n found, check if it satisfies the k component.
if yes, then the assumption is correct and the 3 vectors are indeed coplanar, otherwise …
if no, then the 3 vectors are not coplanar.
eg, c = ma + nb, where m & n are real numbers.
the first part of the question gives us the possible way to solve the second part....
1) we assume that the 3 vectors are coplanar, then c = ma + nb.
2) by comparing the i & j components in the equation, solve for m & n.
3) using the values of m & n found, check if it satisfies the k component.
if yes, then the assumption is correct and the 3 vectors are indeed coplanar, otherwise …
if no, then the 3 vectors are not coplanar.
There is actually a shorter way for part 2 :
Just find the scalar triple product.
Eg.
a.(b x c) or a.(c x b)
b.(c x a) or a.(a x c)
c.(a x b) or a.(b x a)
If the 3 vectors are coplanar, the cross product of 2 of them will be perpendicular to the third vector.
(Recall that the cross product of 2 vectors gives a new vector that is perpendicular to both of them.
And this new vector will therefore be perpendicular to the 3rd vector simce it lies in the same plane as the other two. )
Now, recall that the dot product of 2 perpendicular vectors is 0.
So if you dot the new vector with the 3rd vector, you should get 0 since they are perpendicular.
If your result isn't 0, they arent perpendicular. Which means the three vectors were not coplanar in the first place.
Just find the scalar triple product.
Eg.
a.(b x c) or a.(c x b)
b.(c x a) or a.(a x c)
c.(a x b) or a.(b x a)
If the 3 vectors are coplanar, the cross product of 2 of them will be perpendicular to the third vector.
(Recall that the cross product of 2 vectors gives a new vector that is perpendicular to both of them.
And this new vector will therefore be perpendicular to the 3rd vector simce it lies in the same plane as the other two. )
Now, recall that the dot product of 2 perpendicular vectors is 0.
So if you dot the new vector with the 3rd vector, you should get 0 since they are perpendicular.
If your result isn't 0, they arent perpendicular. Which means the three vectors were not coplanar in the first place.
if I am not wrong, i believe triple products is not in the syllabus, for students taking H2 Maths in 2020 exam.
nevertheless, it is not difficult to understand the concept so perhaps some JCs may still teach it. I think if any student applies a method that is not in the syllabus, and they get the correct answer, they should still get the marks.
nevertheless, it is not difficult to understand the concept so perhaps some JCs may still teach it. I think if any student applies a method that is not in the syllabus, and they get the correct answer, they should still get the marks.
Yes they will be awarded the mark, since the scalar triple product = 0 can be intuitively realised by piecing together the ① concept of cross product of 2 vectors gives a perpendicular vector to the 2 and ② dot product of 2 perpendicular vectors = 0
Hi Sonia,unable to type columns here so here's a substitute :
a = (3 -5 6)
b = (2 3 -2)
c = (-1 8 -8)
(b x c)
(2 3 -2) x (-1 8 -8)
= ( (3 x (-8) - (-2) x 8) ((-2) x (-1) - 2 x (-8)) (2 x 8 - 3 x (-1) )
= (-8 18 19)
Then,
a.(b x c)
= (3 -5 6).(-8 18 19)
= 3 x (-8) + (-5) x 18 + 6 x 19
= -24 - 90 + 114
= 0
Since we get 0, they are coplanar
a = (3 -5 6)
b = (2 3 -2)
c = (-1 8 -8)
(b x c)
(2 3 -2) x (-1 8 -8)
= ( (3 x (-8) - (-2) x 8) ((-2) x (-1) - 2 x (-8)) (2 x 8 - 3 x (-1) )
= (-8 18 19)
Then,
a.(b x c)
= (3 -5 6).(-8 18 19)
= 3 x (-8) + (-5) x 18 + 6 x 19
= -24 - 90 + 114
= 0
Since we get 0, they are coplanar
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Coolant means they lie on the same line so it’s as such
Date Posted:
4 years ago
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