Add and subtract vectors using components

Geometry and measures • Vectors

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What is the effect of rounding components before finding magnitude?

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Early rounding causes small errors in the magnitude and direction of the resultant; delay rounding until the final step.

Key concepts

What you'll likely be quizzed about

Definition of vectors and notation

A vector is a quantity that has both magnitude and direction. Vectors display using boldface (v) or with an arrow above (192) and often appear in component form as (x, y). Vectors differ from scalars because scalars have magnitude only and no direction. Limits on vectors include the need for a common reference direction and compatible units when combining quantities.

Resultant vector and geometric methods

The resultant vector represents the combined effect of two or more vectors. The triangle method places vectors tip-to-tail so that the resultant runs from the tail of the first to the tip of the last, producing the same result as the parallelogram method. The parallelogram method places two vectors tail-to-tail and constructs a parallelogram; the diagonal gives the resultant. Geometric methods provide direct visual cause-effect understanding: combining directions produces a resultant direction and combined magnitudes.

Component method for addition

The component method breaks each vector into horizontal and vertical components. Horizontal components add to give the resultant horizontal component; vertical components add to give the resultant vertical component. The resultant vector follows from these components as (Rx, Ry). Calculating components avoids geometric measurement error and yields precise numerical results for magnitude and direction.

Subtraction as addition of the negative vector

Vector subtraction v 13 w converts to v + (D1w), where (D1w) denotes w reversed in direction but equal in magnitude. Reversing direction changes the sign of each component. Component-wise subtraction follows the rule (v_x - w_x, v_y - w_y). Treating subtraction as addition of the negative ensures consistent application of addition rules and reduces sign errors.

Properties: commutativity and associativity

Vector addition is commutative: v + w = w + v, so order does not change the resultant. Vector addition is associative: (u + v) + w = u + (v + w), so grouping does not change the resultant. Subtraction is not commutative: v 13 w 2 w 13 v in general. Awareness of these properties prevents incorrect reordering of subtraction operations.

Common limitations and checks

Vectors require consistent units and coordinate orientations; mixing coordinate systems or units produces invalid results. Component rounding can cause small numerical errors when finding magnitude and angle. Verification checks include reassembling geometric diagrams, recalculating components, and confirming that vector addition follows commutativity when only addition is involved.

Key notes

Important points to keep in mind

Convert all vectors to the same coordinate axes and units before combining.

Use triangle or parallelogram for visual intuition; use components for precise results.

Perform subtraction as addition of the negative vector to avoid sign mistakes.

Add x-components together and add y-components together to get resultant components.

Find magnitude last using sqrt(R_x^2 + R_y^2) to reduce rounding errors.

Check answers by reversing operations or reconstructing the diagram.

Remember addition is commutative; subtraction is not.

Label axes and clearly mark directions on diagrams to prevent orientation errors.