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junior college 2 | H2 Maths
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Good morning! Can someone pls help me with this question ? Thank u so much and stay healthy eveyone :)
Date Posted: 4 years ago
Views: 262
a = (a.b)b
a.b is the dot product, a scalar product
where a.b =|a||b|cosθ, where θ is the angle between the two vectors.
So this means that means that a and b are
scalar multiples of each other.
They are either parallel (positive multiples of each other, same direction) or antiparallel (opposite directions, negative multiples of each other)
So,
a = (a.b)b = |a||b|cosθ b
If parallel vectors, θ = 0°
Then cos 0° = 1
If antiparallel, θ = 180°
Then cos 180° = -1
so a = |a||b|b or a = -|a||b|b
a /|a| = |b|b or a /|a| = -b|b|
â = |b|b or â = -|b|b
Since â is a unit vector, then |b|b must be a unit vector as well. And it is the unit vector of b.
But b^ is also the unit vector of b.
So b^ = |b|b
b/|b| = |b|b
b = |b|² b
1(b) = |b|² b
Comparing the constants, this means |b|² = 1 and therefore |b| = 1
(Since |b| = 1, then b is actually a unit vector itself. which also means b = b^)
a.b is the dot product, a scalar product
where a.b =|a||b|cosθ, where θ is the angle between the two vectors.
So this means that means that a and b are
scalar multiples of each other.
They are either parallel (positive multiples of each other, same direction) or antiparallel (opposite directions, negative multiples of each other)
So,
a = (a.b)b = |a||b|cosθ b
If parallel vectors, θ = 0°
Then cos 0° = 1
If antiparallel, θ = 180°
Then cos 180° = -1
so a = |a||b|b or a = -|a||b|b
a /|a| = |b|b or a /|a| = -b|b|
â = |b|b or â = -|b|b
Since â is a unit vector, then |b|b must be a unit vector as well. And it is the unit vector of b.
But b^ is also the unit vector of b.
So b^ = |b|b
b/|b| = |b|b
b = |b|² b
1(b) = |b|² b
Comparing the constants, this means |b|² = 1 and therefore |b| = 1
(Since |b| = 1, then b is actually a unit vector itself. which also means b = b^)
Another way to see it :
if â = |b|b or â = -|b|b,
Then |â| = ||b| b | or |-|b| b |
1 = |b||b| = |b|²
So |b| = 1
if â = |b|b or â = -|b|b,
Then |â| = ||b| b | or |-|b| b |
1 = |b||b| = |b|²
So |b| = 1
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Hope you’re able to see :)
1) a•b is a scalar, so basically vector a is just a scalar multiple of vector b. They have to be in the same direction, but provided a•b is positive. Otherwise they are opposite directions.
Vector b has to be a unit vector because Vector a is the vector projection along vector b, which based on the projection formula, Vector b has to be a unit vector.
2) geometrically sketch and show that OP is perpendicular to AP then try to move P around
1) a•b is a scalar, so basically vector a is just a scalar multiple of vector b. They have to be in the same direction, but provided a•b is positive. Otherwise they are opposite directions.
Vector b has to be a unit vector because Vector a is the vector projection along vector b, which based on the projection formula, Vector b has to be a unit vector.
2) geometrically sketch and show that OP is perpendicular to AP then try to move P around
Date Posted:
4 years ago
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