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.

</section> <section style="font-size: 50px";> <h2 id="vectors---scalar-triple-product">Vectors - Scalar triple product</h2> <ul> <li>Introduction to scalar triple product.</li> <li>Formula for the scalar triple product of three vectors.</li> <li>Example: Find the scalar triple product of vectors A = (2, 3, -1), B = (-4, 1, 5), and C = (0, 2, -3).</li> <li>Steps to calculate the scalar triple product.</li> <li>Solution: The scalar triple product of A, B, and C is -30.</li> <li>Interpretation of scalar triple product.</li> <li>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.</li> <li>Solution: Calculate the magnitude of the scalar triple product and compare it with the volume of the parallelepiped.

</section> <section style="font-size: 50px";> <h2 id="vectors---projection-of-a-vector">Vectors - Projection of a vector</h2> <ul> <li>Introduction to projection of a vector.</li> <li>Formula for the projection of vector A onto vector B.</li> <li>Example: Find the projection of vector A = (2, 3) onto vector B = (-4, 1).</li> <li>Steps to calculate the projection.</li> <li>Solution: The projection of A onto B is (-3, 3/2).</li> <li>Properties of vector projection.</li> <li>Example: Show that the projection of A onto B is parallel to B.</li> <li>Solution: Calculate the dot product of the projection vector and B.

</section> <section style="font-size: 50px";> <h2 id="vectors---angle-between-two-vectors">Vectors - Angle between two vectors</h2> <ul> <li>Definition of the angle between two vectors.</li> <li>Formula for calculating the angle between two vectors.</li> <li>Example: Find the angle between vectors A = (2, 3) and B = (-4, 1).</li> <li>Steps to calculate the angle.</li> <li>Solution: The angle between A and B is approximately 133.74 degrees.</li> <li>Properties of the angle between two vectors.</li> <li>Example: Show that the angle between two parallel vectors is 0 degrees.</li> <li>Solution: Calculate the dot product of the vectors and the magnitudes.

</section> <section style="font-size: 50px";> <h2 id="vectors---scalar-and-vector-projection">Vectors - Scalar and vector projection</h2> <ul> <li>Recap of scalar and vector projection.</li> <li>Difference between scalar and vector projection.</li> <li>Geometric interpretation of scalar and vector projection.</li> <li>Example: Find the scalar projection and vector projection of A = (2, 3) onto B = (-4, 1).</li> <li>Steps to calculate the scalar and vector projections.</li> <li>Solution: The scalar projection is -4.92, and the vector projection is (-3.94, 0.98).</li> <li>Applications of scalar and vector projection.</li> <li>Example: Use the scalar and vector projections to find the orthogonal projection of A onto B.</li> <li>Solution: Calculate the orthogonal projection vector using the formula: A - vector projection.

</section> <section style="font-size: 50px";> <h2 id="vectors---the-plane">Vectors - The plane</h2> <ul> <li>Introduction to vectors in a plane.</li> <li>Equation of a plane in 3-dimensional space.</li> <li>General form of the equation of a plane.</li> <li>Example: Write the equation of a plane passing through the points (1, 2, -1), (3, -1, 4), and (-2, 0, 5).</li> <li>Steps to find the equation of a plane.</li> <li>Solution: The equation of the plane is 2x - 3y + 4z + 5 = 0.</li> <li>Intersection of a line and a plane.</li> <li>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.</li> <li>Steps to find the point of intersection.</li> <li>Solution: Substitute the equation of the line into the equation of the plane and solve for t.

</section> <section style="font-size: 50px";> <h2 id="vectors---the-sphere">Vectors - The sphere</h2> <ul> <li>Definition of a sphere.</li> <li>Equation of a sphere in 3-dimensional space.</li> <li>Standard form of the equation of a sphere.</li> <li>Example: Write the equation of a sphere with center (1, -2, 3) and radius 5.</li> <li>Steps to find the equation of a sphere.</li> <li>Solution: The equation of the sphere is (x-1)^2 + (y+2)^2 + (z-3)^2 = 25.</li> <li>Intersection of a line and a sphere.</li> <li>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.</li> <li>Steps to find the points of intersection.</li> <li>Solution: Substitute the equation of the line into the equation of the sphere and solve for t.

</section> <section style="font-size: 50px";> <h2 id="vectors---equation-of-a-line-in-3d">Vectors - Equation of a line in 3D</h2> <ul> <li>Recap of the equation of a line in 2D.</li> <li>Equation of a line in 3-dimensional space.</li> <li>Parametric form of the equation for a line.</li> <li>Example: Write the equation of a line passing through the point (1, 2, -1) and parallel to the vector (2, -1, 3).</li> <li>Steps to find the equation of a line.</li> <li>Solution: The equation of the line is x = 1 + 2t, y = 2 - t, z = -1 + 3t.</li> <li>Intersection of two lines.</li> <li>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).</li> <li>Steps to find the point of intersection.</li> <li>Solution: Equate the coordinates of the two lines and solve for t and s.

</section> <section style="font-size: 50px";> <h2 id="vectors---equation-of-a-plane-in-3d">Vectors - Equation of a plane in 3D</h2> <ul> <li>Recap of the equation of a plane in 2D.</li> <li>Equation of a plane in 3-dimensional space.</li> <li>Parametric form of the equation for a plane.</li> <li>Example: Write the equation of a plane passing through the point (1, 2, -1) and perpendicular to the vector (2, -1, 3).</li> <li>Steps to find the equation of a plane.</li> <li>Solution: The equation of the plane is 2x - y + 3z = 5.</li> <li>Intersection of a line and a plane.</li> <li>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.</li> <li>Steps to find the point of intersection.</li> <li>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.

</section> <section style="font-size: 50px";> <h2 id="vectors---scalar-triple-product-1">Vectors - Scalar triple product</h2> <ul> <li>Recap of scalar triple product.</li> <li>Definition of scalar triple product.</li> <li>Formula for scalar triple product: A • (B x C).</li> <li>Example: Find the scalar triple product of vectors A = (2, 3, -1), B = (-4, 1, 5), and C = (0, 2, -3).</li> <li>Solution: The scalar triple product of A, B, and C is -30.</li> <li>Interpretation of scalar triple product.</li> <li>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.</li> <li>Solution: Calculate the magnitude of the scalar triple product and compare it with the volume of the parallelepiped.</li> <li>Properties of scalar triple product: <ul> <li>It is anti-commutative: A • (B x C) = -(C x B) • A.</li> <li>It is distributive over vector addition: A • (B x C + D x E) = A • (B x C) + A • (D x E).</li> <li>It is not associative: (A • B) x C ≠ A • (B x C).

</section> <section style="font-size: 50px";> <h2 id="vectors---projection-of-a-vector-1">Vectors - Projection of a vector</h2> <ul> <li>Recap of projection of a vector.</li> <li>Definition of vector projection.</li> <li>Formula for vector projection: proj_BA = (A • B / |B|²) B.</li> <li>Example: Find the projection of vector A = (2, 3) onto vector B = (-4, 1).</li> <li>Solution: The projection of A onto B is (-3, 3/2).</li> <li>Properties of vector projection. <ul> <li>The projection of A onto B is parallel to B.</li> <li>The projection of A onto B is orthogonal to the component of A perpendicular to B.</li> <li>The magnitude of the projection vector is ||proj_BA|| = |A| cosθ, where θ is the angle between A and B.</li> </ul> </li> <li>Example: Show that the projection of A onto B is parallel to B.</li> <li>Solution: Calculate the dot product of the projection vector and B.

</section> <section style="font-size: 50px";> <h2 id="vectors---angle-between-two-vectors-1">Vectors - Angle between two vectors</h2> <ul> <li>Recap of angle between vectors.</li> <li>Definition of angle between two vectors.</li> <li>Formula for calculating the angle between two vectors: cosθ = (A • B) / (|A| |B|).</li> <li>Example: Find the angle between vectors A = (2, 3) and B = (-4, 1).</li> <li>Solution: The angle between A and B is approximately 133.74 degrees.</li> <li>Properties of the angle between two vectors. <ul> <li>The angle between two parallel vectors is 0 degrees.</li> <li>The angle between two perpendicular vectors is 90 degrees.</li> <li>The angle between two anti-parallel vectors is 180 degrees.</li> </ul> </li> <li>Example: Show that the angle between two parallel vectors is 0 degrees.</li> <li>Solution: Calculate the dot product of the vectors and the magnitudes.

</section> <section style="font-size: 50px";> <h2 id="vectors---scalar-and-vector-projection-1">Vectors - Scalar and vector projection</h2> <ul> <li>Recap of scalar and vector projection.</li> <li>Difference between scalar and vector projection.</li> <li>Geometric interpretation of scalar and vector projection.</li> <li>Formula for scalar projection: proj_BA = |A| cosθ.</li> <li>Example: Find the scalar projection of A = (2, 3) onto B = (-4, 1).</li> <li>Solution: The scalar projection is -4.92.</li> <li>Formula for vector projection: proj_BA = (A • B / |B|²) B.</li> <li>Example: Find the vector projection of A = (2, 3) onto B = (-4, 1).</li> <li>Solution: The vector projection is (-3.94, 0.98).</li> <li>Applications of scalar and vector projection. <ul> <li>Scalar projection is used to find the component of A in the direction of B.</li> <li>Vector projection is used to find the projection of A onto B.</li> <li>Vector projection can be used to find the orthogonal projection of A onto B.</li> </ul> </li> <li>Example: Use the scalar and vector projections to find the orthogonal projection of A onto B.</li> <li>Solution: Calculate the orthogonal projection vector using the formula: A - vector projection.

</section> <section style="font-size: 50px";> <h2 id="vectors---the-plane-1">Vectors - The plane</h2> <ul> <li>Recap of vectors in a plane.</li> <li>Definition of a plane in 3-dimensional space.</li> <li>Equation of a plane: Ax + By + Cz + D = 0.</li> <li>Example: Write the equation of a plane passing through the points (1, 2, -1), (3, -1, 4), and (-2, 0, 5).</li> <li>Solution: The equation of the plane is 2x - 3y + 4z + 5 = 0.</li> <li>Intersection of a line and a plane.</li> <li>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.</li> <li>Solution: Substitute the equation of the line into the equation of the plane and solve for t.</li> <li>Geometric interpretation of the equation of a plane.</li> <li>Finding the equation of a plane given normal and a point.

</section> <section style="font-size: 50px";> <h2 id="vectors---the-sphere-1">Vectors - The sphere</h2> <ul> <li>Recap of spheres.</li> <li>Definition of a sphere in 3-dimensional space.</li> <li>Equation of a sphere: (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2.</li> <li>Example: Write the equation of a sphere with center (1, -2, 3) and radius 5.</li> <li>Solution: The equation of the sphere is (x-1)^2 + (y+2)^2 + (z-3)^2 = 25.</li> <li>Intersection of a line and a sphere.</li> <li>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.</li> <li>Solution: Substitute the equation of the line into the equation of the sphere and solve for t.</li> <li>Geometric interpretation of the equation of a sphere.</li> <li>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.

</section> <section style="font-size: 50px";> <h2 id="vectors---scalar-triple-product-2">Vectors - Scalar triple product</h2> <ul> <li>Recap of scalar triple product.</li> <li>Definition of scalar triple product.</li> <li>Formula for scalar triple product: A • (B x C).</li> <li>Example: Find the scalar triple product of vectors A = (2, 3, -1), B = (-4, 1, 5), and C = (0, 2, -3).</li> <li>Solution: The scalar triple product of A, B, and C is -30.</li> <li>Interpretation of scalar triple product.</li> <li>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.</li> <li>Solution: Calculate the magnitude of the scalar triple product and compare it with the volume of the parallelepiped.</li> <li>Properties of scalar triple product: <ul> <li>It is anti-commutative: A • (B x C) = -(C x B) • A.</li> <li>It is distributive over vector addition: A • (B x C + D x E) = A • (B x C) + A • (D x E).</li> <li>It is not associative: (A • B) x C ≠ A • (B x C).