3.3 Applications of Vectors

    Cards (68)

    • What are the basic vector operations in vector mathematics?
      Addition, subtraction, scalar multiplication, dot product, cross product
    • The dot product of two vectors results in a scalar
    • The dot product is useful for determining the angle between two vectors.
    • What does the cross product of two vectors in three dimensions produce?
      A perpendicular vector
    • Match the vector operation with its description:
      Addition ↔️ Combines the components of two vectors
      Subtraction ↔️ Subtracts the components of one vector from another
      Scalar Multiplication ↔️ Scales each component of a vector by a scalar
      Dot Product ↔️ Produces a scalar from two vectors
      Cross Product ↔️ Produces a vector perpendicular to two vectors
    • The dot product produces a scalar
    • What is the cross product of two vectors in three dimensions?
      A perpendicular vector
    • The cross product can be used to find the area of a parallelogram formed by two vectors.
    • What are the two forms used to express equations of lines and planes?
      Vector and cartesian
    • The vector equation of a line passing through \vec{a}</latex> in the direction d\vec{d} is r=\vec{r} =a+ \vec{a} +td t\vec{d}, where tt is a scalar parameter
    • What is the cartesian form for a line in 2D space?
      y=y =mx+ mx +c c
    • The vector equation of a plane passing through a\vec{a} with normal vector n\vec{n} is (ra)n=(\vec{r} - \vec{a}) \cdot \vec{n} =0 0.
    • Match the concept with its equation form:
      Line (Vector Form) ↔️ r=\vec{r} =a+ \vec{a} +td t\vec{d}
      Line (Cartesian Form in 2D) ↔️ y=y =mx+ mx +c c
      Plane (Vector Form) ↔️ (ra)n=(\vec{r} - \vec{a}) \cdot \vec{n} =0 0
      Plane (Cartesian Form) ↔️ Ax+Ax +By+ By +Cz= Cz =D D
    • The cartesian form for a plane is Ax + By + Cz = D
    • What is the vector form of a line passing through a point a\vec{a} in the direction d\vec{d}?

      r=\vec{r} =a+ \vec{a} +td t\vec{d}
    • The cartesian form for a line in 2D is y=y =mx+ mx +c c, and in 3D it is \frac{x - x_{0}}{d_{x}} = \frac{y - y_{0}}{d_{y}} = \frac{z - z_{0}}{d_{z}}</latex>
    • What is the vector form of a plane passing through point a\vec{a} with normal vector n\vec{n}?

      (ra)n=(\vec{r} - \vec{a}) \cdot \vec{n} =0 0
    • The cartesian form for a plane is Ax + By + Cz = D</latex>
    • Match the concept with its vector and cartesian forms:
      Line ↔️ r=\vec{r} =a+ \vec{a} +td t\vec{d} ||| xx0dx=\frac{x - x_{0}}{d_{x}} =yy0dy= \frac{y - y_{0}}{d_{y}} =zz0dz \frac{z - z_{0}}{d_{z}}
      Plane ↔️ (ra)n=(\vec{r} - \vec{a}) \cdot \vec{n} =0 0 ||| Ax+Ax +By+ By +Cz= Cz =D D
    • Vectors can be added or subtracted by combining their respective components.
    • The dot product of two vectors results in a scalar</latex>
    • What type of vector results from the cross product of two vectors in three dimensions?
      Perpendicular vector
    • Match the vector operation with its formula:
      Addition ↔️ A+\vec{A} +B= \vec{B} =(a1+ (a_{1} +b1,a2+ b_{1}, a_{2} +b2) b_{2})
      Dot Product ↔️ AB=\vec{A} \cdot \vec{B} =a1b1+ a_{1}b_{1} +a2b2 a_{2}b_{2}
      Cross Product ↔️ A×B=\vec{A} \times \vec{B} =(a2b3a3b2,a3b1a1b3,a1b2a2b1) (a_{2}b_{3} - a_{3}b_{2}, a_{3}b_{1} - a_{1}b_{3}, a_{1}b_{2} - a_{2}b_{1})
    • What is the vector equation of a line passing through point a\vec{a} in the direction d\vec{d}?

      r=\vec{r} =a+ \vec{a} +td t\vec{d}
    • What is the vector equation of a plane passing through point a\vec{a} with normal vector n\vec{n}?

      (ra)n=(\vec{r} - \vec{a}) \cdot \vec{n} =0 0
    • The vector equation of a line passing through point a\vec{a} in the direction d\vec{d} is \vec{r} = \vec{a} + t\vec{d}</latex>
    • The vector equation of a plane passing through point a\vec{a} with normal vector n\vec{n} is (\vec{r} - \vec{a}) \cdot \vec{n} = 0</latex>
    • What is the scalar parameter in the vector equation of a line?
      tt
    • What is the Cartesian form of the vector equation of a line?
      xx0dx=\frac{x - x_{0}}{d_{x}} =yy0dy= \frac{y - y_{0}}{d_{y}} =zz0dz \frac{z - z_{0}}{d_{z}}
    • What is the Cartesian form of the vector equation of a line?
      xx0dx=\frac{x - x_{0}}{d_{x}} =yy0dy= \frac{y - y_{0}}{d_{y}} =zz0dz \frac{z - z_{0}}{d_{z}}
    • The Cartesian form of the vector equation of a plane is Ax+Ax +By+ By +Cz= Cz =D D
    • What is the purpose of the vector equations of lines and planes?
      Representing lines and planes
    • Match the scenario with its distance formula:
      Two points ↔️ d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}</latex>
      Point and line ↔️ d=d =AP×dd \frac{|\vec{AP} \times \vec{d}|}{|\vec{d}|}
      Point and plane ↔️ d=d = \frac{|Ax_{0} +By0+ By_{0} + Cz_{0} - D|}{\sqrt{A^{2} +B2+ B^{2} + C^{2}}}
      Parallel lines ↔️ d=d =AB×dd \frac{|\vec{AB} \times \vec{d}|}{|\vec{d}|}
      Parallel planes ↔️ d=d = \frac{|D_{2} - D_{1}|}{\sqrt{A^{2} +B2+ B^{2} + C^{2}}}
    • What is the formula for the distance between two points A(x1,y1,z1)A(x_{1}, y_{1}, z_{1}) and B(x2,y2,z2)B(x_{2}, y_{2}, z_{2})?

      d=d = \sqrt{(x_{2} - x_{1})^{2} +(y2y1)2+ (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}
    • The distance between a point P(x0,y0,z0)P(x_{0}, y_{0}, z_{0}) and a plane Ax+Ax +By+ By +Cz= Cz =D D is d=d = \frac{|Ax_{0} +By0+ By_{0} + Cz_{0} - D|}{\sqrt{A^{2} +B2+ B^{2} + C^{2}}}
    • What is the formula for the distance between two parallel lines r=\vec{r} =a+ \vec{a} +td t\vec{d} and r=\vec{r} =b+ \vec{b} +td t\vec{d}?

      d=d =AB×dd \frac{|\vec{AB} \times \vec{d}|}{|\vec{d}|}
    • Match the scenario with its distance formula:
      Two points ↔️ d=d = \sqrt{(x_{2} - x_{1})^{2} +(y2y1)2+ (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}
      Point and line ↔️ d=d =AP×dd \frac{|\vec{AP} \times \vec{d}|}{|\vec{d}|}
      Point and plane ↔️ d=d = \frac{|Ax_{0} +By0+ By_{0} + Cz_{0} - D|}{\sqrt{A^{2} +B2+ B^{2} + C^{2}}}
      Parallel lines ↔️ d=d =AB×dd \frac{|\vec{AB} \times \vec{d}|}{|\vec{d}|}
      Parallel planes ↔️ d=d = \frac{|D_{2} - D_{1}|}{\sqrt{A^{2} +B2+ B^{2} + C^{2}}}
    • The distance between two parallel planes Ax+Ax +By+ By +Cz= Cz =D1 D_{1} and Ax+Ax +By+ By +Cz= Cz =D2 D_{2} is given by d=d = \frac{|D_{2} - D_{1}|}{\sqrt{A^{2} +B2+ B^{2} + C^{2}}}, where d represents the distance.
    • The distance formula between two points in 3D space is d=d = \sqrt{(x_{2} - x_{1})^{2} +(y2y1)2+ (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}.
    • The distance between a point and a line is given by d = \frac{|\vec{AP} \times \vec{d}|}{|\vec{d}|}</latex>, where AP is a vector from the point to the line.