A projection may refer to a signed scalar component or a vector along a specified direction. Check which object the question requests before using a formula.
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Calculate the dot product
Let a = (3, 4) and b = (1, 2). Their dot product is 11, while b has squared length 5 and length √5.
The scalar projection of a along b is 11/√5. The vector projection is (11/5)b = (11/5, 22/5). The scalar alone does not contain the direction vector.
Check the perpendicular remainder
Subtract the projection from a to obtain (4/5, -2/5). Its dot product with b is 4/5 - 4/5 = 0, confirming that the residual is perpendicular to b.
The projection and residual add back to the original vector, providing a second check on the components.
Keep the direction vector nonzero
A projection onto the zero vector's direction is not defined by these formulas, because its length is zero and it has no direction. This is a condition, not a computational inconvenience to ignore.
This original exercise develops algebraic geometry reasoning for AMC 12 preparation, not an official topic prediction. Label scalar and vector answers clearly, and use MAA resources for current competition information.
Continue reading: AMC 12 Trigonometry: Solve Equations Without Losing Solutions.
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Official information and further reading: Source 1
