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Vectors: components, magnitude and unit vectors
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A vector has size and direction. Writing vectors with i and j turns geometry into simple arithmetic.
i and j
i is one unit in the x direction and j is one unit in the y direction.
a = 2i + 3j and b = i โ 4j
Add and subtract the parts separately: 2a โ b = (4 โ 1)i + (6 + 4)j = 3i + 10j
a = 2i + 3j and b = i โ 4j
Add and subtract the parts separately: 2a โ b = (4 โ 1)i + (6 + 4)j = 3i + 10j
Magnitude and direction
|ai + bj| = โ(a2 + b2)
|3i + 4j| = 5
The angle with i is found with tan: tan ฮธ = 43, so ฮธ = 53.1ยฐ.
A unit vector has magnitude 1: divide by the magnitude. 15(3i + 4j) = 0.6i + 0.8j
|3i + 4j| = 5
The angle with i is found with tan: tan ฮธ = 43, so ฮธ = 53.1ยฐ.
A unit vector has magnitude 1: divide by the magnitude. 15(3i + 4j) = 0.6i + 0.8j
Position vectors
The position vector of A is OA, the vector from the origin O.
AB = OB โ OA, often written b โ a.
A(1, 2), B(4, โ2): AB = (4 โ 1)i + (โ2 โ 2)j = 3i โ 4j
AB = OB โ OA, often written b โ a.
A(1, 2), B(4, โ2): AB = (4 โ 1)i + (โ2 โ 2)j = 3i โ 4j
Find a unit vector in the direction of 6i + 8j.
- |6i + 8j| = โ(36 + 64) = 10
- Divide each part by 10
Answer: 0.6i + 0.8j
Work with the i and j parts separately. |ai + bj| = โ(a2 + b2). AB = b โ a. Unit vector = vector รท magnitude.
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