Are (3,4) and (4,−3) perpendicular

Given the coordinates of planar vector a are (3, 4), and the coordinates of vector b are (4, -3), find their dot product and determine whether the two vectors are perpendicular.

Step-by-step

1. a = (3, 4), b = (4, -3)
   First write both vectors in coordinate form and identify corresponding components.
2. a·b = x₁x₂ + y₁y₂ = 3 × 4 + 4 × -3 = 0
   The dot product in coordinates equals the sum of products of corresponding components; the result is a number, not a vector.
3. |a| = √(3² + 4²) = 5, |b| = 5
   The magnitude of a vector is the principal square root of the sum of squares of its components.
4. cosθ = a·b/(|a||b|) = 0 ⇒ θ = 90°
   Cosine of the included angle equals the dot product divided by the product of the two magnitudes.
5. a · b = 0 ⇒ the two vectors are perpendicular
   A dot product of zero is necessary and sufficient for perpendicularity; it can be used to quickly judge orthogonality.

Why it works

A dot product of zero is equivalent to two vectors being perpendicular; this is a necessary and sufficient condition, so there is no need to calculate the angle to judge perpendicularity. In this problem, the components of the two vectors are exactly swapped with one sign difference, which is a typical feature of a ninety-degree rotation, and calculating a dot product of zero confirms this. Except for the zero vector, whose dot product with any other vector is always zero but has no direction.

Common mistakes

• 1) Calculating ninety degrees with the angle formula and thinking you must write an arccosine, adding an unnecessary step that is prone to error.
• 2) Stating "neither vector exists" for a dot product of zero, distorting the conclusion.
• 3) Incorrectly handling signs when multiplying components, dropping one negative sign and calculating the dot product as 24.

Related

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