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This printable supports Common Core Mathematics Standard HSN-VM.B.5, HSN-VM.B.5b

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Scalar Multiplication, Magnitude, & Direction (Grades 11-12)

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Scalar Multiplication, Magnitude, & Direction

Instructions: Unless otherwise stated, the direction of a vector is considered to be the angle between the positive x-axis and the vector (with its tail at the origin), going from the positive x-axis to the vector in a counterclockwise direction.

1. 
For [math]vec{v} = <<-4,2>>[/math], find [math]||3vec{v}||[/math].
  1. [math]6[/math]
  2. [math]2sqrt(5)[/math]
  3. [math]6sqrt(5)[/math]
  4. [math]60[/math]
2. 
If [math]||5vec{u}|| = 15[/math], what is the value of [math]||vec{u}|| ?[/math]
  1. 3
  2. 5
  3. 10
  4. Not enough information to determine.
3. 
If the direction of [math]vec{v}[/math] is 60°, what is the direction of [math]-3vec{v}[/math]?
  1. 240°
  2. -180°
  3. 60°
  4. 180°
4. 
For vectors [math]vec{n} = 6vec{v}[/math] and [math]vec{m} = -12vec{v}[/math], which of the following statements are correct? Choose all correct answers.
  1. The vectors have opposite directions.
  2. [math]-2||vec{n}|| = ||vec{m}||[/math]
  3. Both vectors are in the same direction as [math]vec{v}[/math].
  4. The magnitude of [math]vec{m}[/math] is twice that of [math]vec{n}[/math].
5. 
For [math]vec{v} = <<0,4>>[/math], find the direction of [math]3vec{v}[/math].
  1. 270°
  2. 180°
  3. 90°
6. 
For [math]vec{v} = <<3,2>>[/math], find [math]|| -10vec{v}||[/math].
  1. [math] -10sqrt{13}[/math]
  2. [math]sqrt{130}[/math]
  3. [math]50[/math]
  4. [math]10sqrt{13}[/math]
7. 
If you are given [math]vec{v}[/math], which of the following is a vector that is twice as long but in the opposite direction?
  1. [math]2vec{v}[/math]
  2. [math]-2vec{v}[/math]
  3. [math]1/2 vec{v}[/math]
  4. [math]-1/2 vec{v}[/math]
8. 
For a given vector, [math]vec{v}[/math], and scalar multiple [math]a, \ a<0[/math], which of the following statements is correct?
  1. The direction of [math]a \ vec{v}[/math] is the same as the direction of [math]vec{v}[/math].
  2. The direction of [math]a \ vec{v}[/math] is opposite to that of [math]vec{v}[/math].
  3. The direction of the two vectors is different, but it is impossible to tell how they are different.
  4. The direction of the two vectors is different, but the difference will be less than 180°.
9. 
Which of the following statements is true if [math]vec{v}[/math] is a non-zero vector, [math]a[/math] is a scalar such that [math]a< -1[/math], and [math]vec{w} = a vec{v} ?[/math]
  1. [math]||vec{w}|| = ||a vec{v}||[/math]
  2. [math]||vec{w}|| < ||a vec{v} || [/math]
  3. [math]||vec{w}|| > ||a vec{v} || [/math]
  4. [math] || vec{w}|| = - || vec{v}||[/math]
10. 
For the vectors [math]vec{n}[/math] and [math]vec{m}[/math], [math]||a \ vec{n}|| = ||vec{m}||[/math] for scalar [math]a[/math], where [math]a[/math] can be any non-zero real number. Which of the following statements must be true?
  1. The two vectors have the same direction.
  2. The two vectors have the opposite direction.
  3. The ratio of the magnitudes of [math]vec{m}[/math] to [math]vec{n}[/math] is [math]a[/math] to [math]1[/math].
  4. The two vectors are equal.

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