Or press and hold the picture and choose “Save image”.
for Whoobee's shop
Vectors: Direction, Unit Vectors & Angles — Easy Worksheet 1
Grade 12 · MathSheets Name: ____________________ Date: __________
Give exact answers. Write vectors as <a, b> and angles in degrees (0° ≤ θ < 360° for direction angles).
How to solve
The angle between u and v satisfies cos θ = u · v|u||v|. A vector with magnitude m and direction θ is <m cos θ, m sin θ>; the unit vector in the direction of v is v / |v|. Two vectors are perpendicular when u · v = 0.
Example
u = <0, -1> and v = <-5, 1>. Find 3u - v.
Answer: <5, -4>
Not quite. The answer is
💡 Multiply each vector by its number: 3u = <3 × (-1), 3 × (-5)> = <-3, -15>, and -2v = <-2 × (-5), -2 × 2> = <10, -4>. Add the matching components: <-3 + 10, -15 + (-4)> = <7, -19>.
✓ Correct!
Not quite. The answer is
💡 Multiply each vector by its number: 3u = <3 × (-1), 3 × (-5)> = <-3, -15>, and -2v = <-2 × (-5), -2 × 2> = <10, -4>. Add the matching components: <-3 + 10, -15 + (-4)> = <7, -19>.
💡 How to solve: Vectors: Direction, Unit Vectors & Angles
The angle between u and v satisfies cos θ = u · v|u||v|. A vector with magnitude m and direction θ is <m cos θ, m sin θ>; the unit vector in the direction of v is v / |v|. Two vectors are perpendicular when u · v = 0.
Try a similar one
u = <0, 4> and v = <-5, 1>. Find 3u + v.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 First find the magnitude: |v| = √((-4)2 + 32) = √(16 + 9) = √25 = 5. Divide each component by 5: <-45, 35>. The unit vector is <-45, 35>.
✓ Correct!
Not quite. The answer is
💡 First find the magnitude: |v| = √((-4)2 + 32) = √(16 + 9) = √25 = 5. Divide each component by 5: <-45, 35>. The unit vector is <-45, 35>.
💡 How to solve: Vectors: Direction, Unit Vectors & Angles
The angle between u and v satisfies cos θ = u · v|u||v|. A vector with magnitude m and direction θ is <m cos θ, m sin θ>; the unit vector in the direction of v is v / |v|. Two vectors are perpendicular when u · v = 0.
Try a similar one
u = <-5, -1> and v = <-1, -4>. Find 4u - v.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use v = <|v| cos θ, |v| sin θ>. cos 60° = 12 and sin 60° = √32, so v = <12 × 12, 12 × √32> = <6, 6√3>.
✓ Correct!
Not quite. The answer is
💡 Use v = <|v| cos θ, |v| sin θ>. cos 60° = 12 and sin 60° = √32, so v = <12 × 12, 12 × √32> = <6, 6√3>.
💡 How to solve: Vectors: Direction, Unit Vectors & Angles
The angle between u and v satisfies cos θ = u · v|u||v|. A vector with magnitude m and direction θ is <m cos θ, m sin θ>; the unit vector in the direction of v is v / |v|. Two vectors are perpendicular when u · v = 0.
Try a similar one
u = <-3, -3> and v = <0, 5>. Find 2u - 3v.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 v = <-3, 0> points straight along the negative x-axis, so its direction angle is 180°.
✓ Correct!
Not quite. The answer is
💡 v = <-3, 0> points straight along the negative x-axis, so its direction angle is 180°.
💡 How to solve: Vectors: Direction, Unit Vectors & Angles
The angle between u and v satisfies cos θ = u · v|u||v|. A vector with magnitude m and direction θ is <m cos θ, m sin θ>; the unit vector in the direction of v is v / |v|. Two vectors are perpendicular when u · v = 0.
Try a similar one
u = <-2, 1> and v = <3, 5>. Find 2u - 2v.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The reference angle θ' has tan θ' = |y| / |x| = 22 = 1, so θ' = 45°. x is positive and y is negative, so v is in Quadrant IV: θ = 360° − 45° = 315°.
✓ Correct!
Not quite. The answer is
💡 The reference angle θ' has tan θ' = |y| / |x| = 22 = 1, so θ' = 45°. x is positive and y is negative, so v is in Quadrant IV: θ = 360° − 45° = 315°.
💡 How to solve: Vectors: Direction, Unit Vectors & Angles
The angle between u and v satisfies cos θ = u · v|u||v|. A vector with magnitude m and direction θ is <m cos θ, m sin θ>; the unit vector in the direction of v is v / |v|. Two vectors are perpendicular when u · v = 0.
Try a similar one
u = <-2, 1> and v = <-5, -4>. Find 3u - 3v.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Multiply each vector by its number: 3u = <3 × 0, 3 × 4> = <0, 12>, and 3v = <3 × 5, 3 × (-1)> = <15, -3>. Add the matching components: <0 + 15, 12 + (-3)> = <15, 9>.
✓ Correct!
Not quite. The answer is
💡 Multiply each vector by its number: 3u = <3 × 0, 3 × 4> = <0, 12>, and 3v = <3 × 5, 3 × (-1)> = <15, -3>. Add the matching components: <0 + 15, 12 + (-3)> = <15, 9>.
💡 How to solve: Vectors: Direction, Unit Vectors & Angles
The angle between u and v satisfies cos θ = u · v|u||v|. A vector with magnitude m and direction θ is <m cos θ, m sin θ>; the unit vector in the direction of v is v / |v|. Two vectors are perpendicular when u · v = 0.
Try a similar one
u = <4, -2> and v = <-4, -2>. Find 2u + v.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 First find the magnitude: |v| = √(72 + 242) = √(49 + 576) = √625 = 25. Divide each component by 25: <725, 2425>. The unit vector is <725, 2425>.
✓ Correct!
Not quite. The answer is
💡 First find the magnitude: |v| = √(72 + 242) = √(49 + 576) = √625 = 25. Divide each component by 25: <725, 2425>. The unit vector is <725, 2425>.
💡 How to solve: Vectors: Direction, Unit Vectors & Angles
The angle between u and v satisfies cos θ = u · v|u||v|. A vector with magnitude m and direction θ is <m cos θ, m sin θ>; the unit vector in the direction of v is v / |v|. Two vectors are perpendicular when u · v = 0.
Try a similar one
u = <5, 0> and v = <5, -2>. Find 3u - 3v.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Multiply each vector by its number: 2u = <2 × 1, 2 × 3> = <2, 6>, and 3v = <3 × (-4), 3 × 1> = <-12, 3>. Add the matching components: <2 + (-12), 6 + 3> = <-10, 9>.
✓ Correct!
Not quite. The answer is
💡 Multiply each vector by its number: 2u = <2 × 1, 2 × 3> = <2, 6>, and 3v = <3 × (-4), 3 × 1> = <-12, 3>. Add the matching components: <2 + (-12), 6 + 3> = <-10, 9>.
💡 How to solve: Vectors: Direction, Unit Vectors & Angles
The angle between u and v satisfies cos θ = u · v|u||v|. A vector with magnitude m and direction θ is <m cos θ, m sin θ>; the unit vector in the direction of v is v / |v|. Two vectors are perpendicular when u · v = 0.
Try a similar one
u = <0, 5> and v = <-4, -5>. Find 2u - 2v.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Log in to save your progress.
Vectors: Direction, Unit Vectors & Angles — Easy Worksheet 1
Give exact answers. Write vectors as <a, b> and angles in degrees (0° ≤ θ < 360° for direction angles).