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Precalculus practice

Vectors in the Plane worksheets

Find components, magnitude, direction, unit vectors, operations, displacement resultants, and dot products in two dimensions.

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Coordinate descriptions carry restrictions

These sheets connect parametric, rectangular, polar, and vector representations while keeping the interval, orientation, quadrant, or algebraic restriction that makes each conversion equivalent.

Learning objective

What this activity builds

Represent and combine 2D vectors exactly, then use dot products to test perpendicularity or find an included angle.

Worked example

Find the magnitude and unit vector for <3,4>.

Magnitude 5; unit vector <3/5,4/5>.

The magnitude is sqrt(3²+4²), and dividing each component by 5 produces length 1.

Teaching notes

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  • Subtract terminal minus initial coordinates.
  • Use atan2 or quadrant reasoning for direction.
  • The zero vector has no direction and is excluded.

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Choose 2D vector skills, notation, value profiles, and displacement applications.

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Math SheetNameScoreTeacherDateVectors in the PlaneWork in two dimensions. Give exact magnitudes and angles when possible, and show vector operations component by component.1.Treat u and v as successive displacements. u = <9, -3> and v = -1i + 0.5j; find u + v, u - v, and 2u.2.Find the resultant u + v for u = 1.5i + 2j and v = <-2, 2>.Use exact values unless a decimal angle is requested.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateVectors in the PlaneWork in two dimensions. Give exact magnitudes and angles when possible, and show vector operations component by component.3.Two displacement directions are u = <4, -5> and v = -1i + 1.5j. Compute u dot v and decide whether they are perpendicular; if not, give theincluded angle to the nearest hundredth of a degree.4.Find a unit vector in the direction of v = 2.5i + 3j.Use exact values unless a decimal angle is requested.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateVectors in the PlaneWork in two dimensions. Give exact magnitudes and angles when possible, and show vector operations component by component.5.Find the exact magnitude and direction angle (0° <= theta < 360°) of the displacement = <6, -2>. Round a nonstandard direction angle to thenearest hundredth of a degree.6.A vector starts at P(-1, 1) and ends at Q(2.5, 2.5). Write it in both component and i, j form.Use exact values unless a decimal angle is requested.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateVectors in the PlaneWork in two dimensions. Give exact magnitudes and angles when possible, and show vector operations component by component.7.Two displacement directions are u = <8, -4> and v = -1i + 0.5j. Compute u dot v and decide whether they are perpendicular; if not, give theincluded angle to the nearest hundredth of a degree.8.A vector starts at P(1, 0) and ends at Q(5.5, 2.5). Write it in both component and i, j form.Use exact values unless a decimal angle is requested.math-sheet.comPage 4 of 4

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