Vectors - Analytic expression of cross product
- Recall the definition of a vector.
- Introduction to cross product.
- Cross product of two vectors in terms of their components.
- Formula for the cross product of two vectors in 3-dimensional space.
- Example: Find the cross product of vectors A = (2, 3, -1) and B = (-4, 1, 5).
- Steps to calculate the cross product.
- Solution: The cross product of A and B is (16, -18, 14).
- Properties of cross product.
- Example: Show that A x B is orthogonal to both A and B.
- Solution: Calculate the dot product of A x B with A and B separately.
Vectors - Scalar triple product
- Introduction to scalar triple product.
- Formula for the scalar triple product of three vectors.
- Example: Find the scalar triple product of vectors A = (2, 3, -1), B = (-4, 1, 5), and C = (0, 2, -3).
- Steps to calculate the scalar triple product.
- Solution: The scalar triple product of A, B, and C is -30.
- Interpretation of scalar triple product.
- Example: Show that the volume of a parallelepiped formed by A, B, and C is given by the absolute value of their scalar triple product.
- Solution: Calculate the magnitude of the scalar triple product and compare it with the volume of the parallelepiped.
Vectors - Projection of a vector
- Introduction to projection of a vector.
- Formula for the projection of vector A onto vector B.
- Example: Find the projection of vector A = (2, 3) onto vector B = (-4, 1).
- Steps to calculate the projection.
- Solution: The projection of A onto B is (-3, 3/2).
- Properties of vector projection.
- Example: Show that the projection of A onto B is parallel to B.
- Solution: Calculate the dot product of the projection vector and B.
Vectors - Angle between two vectors
- Definition of the angle between two vectors.
- Formula for calculating the angle between two vectors.
- Example: Find the angle between vectors A = (2, 3) and B = (-4, 1).
- Steps to calculate the angle.
- Solution: The angle between A and B is approximately 133.74 degrees.
- Properties of the angle between two vectors.
- Example: Show that the angle between two parallel vectors is 0 degrees.
- Solution: Calculate the dot product of the vectors and the magnitudes.
Vectors - Scalar and vector projection
- Recap of scalar and vector projection.
- Difference between scalar and vector projection.
- Geometric interpretation of scalar and vector projection.
- Example: Find the scalar projection and vector projection of A = (2, 3) onto B = (-4, 1).
- Steps to calculate the scalar and vector projections.
- Solution: The scalar projection is -4.92, and the vector projection is (-3.94, 0.98).
- Applications of scalar and vector projection.
- Example: Use the scalar and vector projections to find the orthogonal projection of A onto B.
- Solution: Calculate the orthogonal projection vector using the formula: A - vector projection.
Vectors - The plane
- Introduction to vectors in a plane.
- Equation of a plane in 3-dimensional space.
- General form of the equation of a plane.
- Example: Write the equation of a plane passing through the points (1, 2, -1), (3, -1, 4), and (-2, 0, 5).
- Steps to find the equation of a plane.
- Solution: The equation of the plane is 2x - 3y + 4z + 5 = 0.
- Intersection of a line and a plane.
- Example: Find the point of intersection of the line given by r = (2, 1, 3) + t(1, -2, 1) and the plane 2x - 3y + 4z + 5 = 0.
- Steps to find the point of intersection.
- Solution: Substitute the equation of the line into the equation of the plane and solve for t.
Vectors - The sphere
- Definition of a sphere.
- Equation of a sphere in 3-dimensional space.
- Standard form of the equation of a sphere.
- Example: Write the equation of a sphere with center (1, -2, 3) and radius 5.
- Steps to find the equation of a sphere.
- Solution: The equation of the sphere is (x-1)^2 + (y+2)^2 + (z-3)^2 = 25.
- Intersection of a line and a sphere.
- Example: Find the points of intersection of the line given by r = (2, 1, 3) + t(1, -2, 1) and the sphere (x-1)^2 + (y+2)^2 + (z-3)^2 = 25.
- Steps to find the points of intersection.
- Solution: Substitute the equation of the line into the equation of the sphere and solve for t.
Vectors - Equation of a line in 3D
- Recap of the equation of a line in 2D.
- Equation of a line in 3-dimensional space.
- Parametric form of the equation for a line.
- Example: Write the equation of a line passing through the point (1, 2, -1) and parallel to the vector (2, -1, 3).
- Steps to find the equation of a line.
- Solution: The equation of the line is x = 1 + 2t, y = 2 - t, z = -1 + 3t.
- Intersection of two lines.
- Example: Find the point of intersection of the lines given by r = (1, 2, -1) + t(2, -1, 3) and r = (-1, 0, 3) + s(3, 2, -1).
- Steps to find the point of intersection.
- Solution: Equate the coordinates of the two lines and solve for t and s.
Vectors - Equation of a plane in 3D
- Recap of the equation of a plane in 2D.
- Equation of a plane in 3-dimensional space.
- Parametric form of the equation for a plane.
- Example: Write the equation of a plane passing through the point (1, 2, -1) and perpendicular to the vector (2, -1, 3).
- Steps to find the equation of a plane.
- Solution: The equation of the plane is 2x - y + 3z = 5.
- Intersection of a line and a plane.
- Example: Find the point of intersection of the line given by r = (1, 2, -1) + t(2, -1, 3) and the plane 2x - y + 3z = 5.
- Steps to find the point of intersection.
- Solution: Substitute the equation of the line into the equation of the plane and solve for t.
Vectors - Analytic expression of cross product
- Recap of cross product.
- Geometric interpretation of cross product.
- Formula for cross product in terms of components: A x B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx).
- Example: Find the cross product of vectors A = (2, 3, -1) and B = (-4, 1, 5).
- Solution: The cross product of A and B is (16, -18, 14).
- Properties of cross product.
- It is anti-commutative: A x B = -(B x A).
- It distributes over vector addition: A x (B + C) = A x B + A x C.
- It is not associative: (A x B) x C ≠ A x (B x C).
- The cross product of two parallel vectors is the zero vector.
- Example: Show that the cross product of A and B is orthogonal to both A and B.
- Solution: Calculate the dot product of A x B with A and B separately.
- The magnitude of the cross product corresponds to the area of the parallelogram formed by A and B.
Vectors - Scalar triple product
- Recap of scalar triple product.
- Definition of scalar triple product.
- Formula for scalar triple product: A • (B x C).
- Example: Find the scalar triple product of vectors A = (2, 3, -1), B = (-4, 1, 5), and C = (0, 2, -3).
- Solution: The scalar triple product of A, B, and C is -30.
- Interpretation of scalar triple product.
- Example: Show that the volume of a parallelepiped formed by A, B, and C is given by the absolute value of their scalar triple product.
- Solution: Calculate the magnitude of the scalar triple product and compare it with the volume of the parallelepiped.
- Properties of scalar triple product:
- It is anti-commutative: A • (B x C) = -(C x B) • A.
- It is distributive over vector addition: A • (B x C + D x E) = A • (B x C) + A • (D x E).
- It is not associative: (A • B) x C ≠ A • (B x C).
Vectors - Projection of a vector
- Recap of projection of a vector.
- Definition of vector projection.
- Formula for vector projection: projBA = (A • B / |B|²) B.
- Example: Find the projection of vector A = (2, 3) onto vector B = (-4, 1).
- Solution: The projection of A onto B is (-3, 3/2).
- Properties of vector projection.
- The projection of A onto B is parallel to B.
- The projection of A onto B is orthogonal to the component of A perpendicular to B.
- The magnitude of the projection vector is ||projBA|| = |A| cosθ, where θ is the angle between A and B.
- Example: Show that the projection of A onto B is parallel to B.
- Solution: Calculate the dot product of the projection vector and B.
Vectors - Angle between two vectors
- Recap of angle between vectors.
- Definition of angle between two vectors.
- Formula for calculating the angle between two vectors: cosθ = (A • B) / (|A| |B|).
- Example: Find the angle between vectors A = (2, 3) and B = (-4, 1).
- Solution: The angle between A and B is approximately 133.74 degrees.
- Properties of the angle between two vectors.
- The angle between two parallel vectors is 0 degrees.
- The angle between two perpendicular vectors is 90 degrees.
- The angle between two anti-parallel vectors is 180 degrees.
- Example: Show that the angle between two parallel vectors is 0 degrees.
- Solution: Calculate the dot product of the vectors and the magnitudes.
Vectors - Scalar and vector projection
- Recap of scalar and vector projection.
- Difference between scalar and vector projection.
- Geometric interpretation of scalar and vector projection.
- Formula for scalar projection: projBA = |A| cosθ.
- Example: Find the scalar projection of A = (2, 3) onto B = (-4, 1).
- Solution: The scalar projection is -4.92.
- Formula for vector projection: projBA = (A • B / |B|²) B.
- Example: Find the vector projection of A = (2, 3) onto B = (-4, 1).
- Solution: The vector projection is (-3.94, 0.98).
- Applications of scalar and vector projection.
- Scalar projection is used to find the component of A in the direction of B.
- Vector projection is used to find the projection of A onto B.
- Vector projection can be used to find the orthogonal projection of A onto B.
- Example: Use the scalar and vector projections to find the orthogonal projection of A onto B.
- Solution: Calculate the orthogonal projection vector using the formula: A - vector projection.
Vectors - The plane
- Recap of vectors in a plane.
- Definition of a plane in 3-dimensional space.
- Equation of a plane: Ax + By + Cz + D = 0.
- Example: Write the equation of a plane passing through the points (1, 2, -1), (3, -1, 4), and (-2, 0, 5).
- Solution: The equation of the plane is 2x - 3y + 4z + 5 = 0.
- Intersection of a line and a plane.
- Example: Find the point of intersection of the line given by r = (2, 1, 3) + t(1, -2, 1) and the plane 2x - 3y + 4z + 5 = 0.
- Solution: Substitute the equation of the line into the equation of the plane and solve for t.
- Geometric interpretation of the equation of a plane.
- Finding the equation of a plane given normal and a point.
Vectors - The sphere
- Recap of spheres.
- Definition of a sphere in 3-dimensional space.
- Equation of a sphere: (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2.
- Example: Write the equation of a sphere with center (1, -2, 3) and radius 5.
- Solution: The equation of the sphere is (x-1)^2 + (y+2)^2 + (z-3)^2 = 25.
- Intersection of a line and a sphere.
- Example: Find the points of intersection of the line given by r = (2, 1, 3) + t(1, -2, 1) and the sphere (x-1)^2 + (y+2)^2 + (z-3)^2 = 25.
- Solution: Substitute the equation of the line into the equation of the sphere and solve for t.
- Geometric interpretation of the equation of a sphere.
- Finding the equation of a sphere given center and a point.
Vectors - Analytic expression of cross product
- Recap of cross product.
- Geometric interpretation of cross product.
- Formula for cross product in terms of components: A x B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx).
- Example: Find the cross product of vectors A = (2, 3, -1) and B = (-4, 1, 5).
- Solution: The cross product of A and B is (16, -18, 14).
- Properties of cross product.
- It is anti-commutative: A x B = -(B x A).
- It distributes over vector addition: A x (B + C) = A x B + A x C.
- It is not associative: (A x B) x C ≠ A x (B x C).
- The cross product of two parallel vectors is the zero vector.
- Example: Show that the cross product of A and B is orthogonal to both A and B.
- Solution: Calculate the dot product of A x B with A and B separately.
- The magnitude of the cross product corresponds to the area of the parallelogram formed by A and B.
Vectors - Scalar triple product
- Recap of scalar triple product.
- Definition of scalar triple product.
- Formula for scalar triple product: A • (B x C).
- Example: Find the scalar triple product of vectors A = (2, 3, -1), B = (-4, 1, 5), and C = (0, 2, -3).
- Solution: The scalar triple product of A, B, and C is -30.
- Interpretation of scalar triple product.
- Example: Show that the volume of a parallelepiped formed by A, B, and C is given by the absolute value of their scalar triple product.
- Solution: Calculate the magnitude of the scalar triple product and compare it with the volume of the parallelepiped.
- Properties of scalar triple product:
- It is anti-commutative: A • (B x C) = -(C x B) • A.
- It is distributive over vector addition: A • (B x C + D x E) = A • (B x C) + A • (D x E).
- It is not associative: (A • B) x C ≠ A • (B x C).
Vectors - Analytic expression of cross product Recall the definition of a vector. Introduction to cross product. Cross product of two vectors in terms of their components. Formula for the cross product of two vectors in 3-dimensional space. Example: Find the cross product of vectors A = (2, 3, -1) and B = (-4, 1, 5). Steps to calculate the cross product. Solution: The cross product of A and B is (16, -18, 14). Properties of cross product. Example: Show that A x B is orthogonal to both A and B. Solution: Calculate the dot product of A x B with A and B separately.