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Section Formula

Internal division Let A and B be two points with position vectors 79290.png respectively, and C be a point dividing AB internally in the ratio m : n. Then the position vector of C is given by
External division Let A and B be two points with position vectors 79305.png respectively and let C be a point dividing 79311.png externally in the ratio m : n. Then the position vector of 79317.png is given by

Addition of two vectors

Let 79332.png and 79341.png be any two vectors.
Then their sum or resultant, denoted by 79351.pngis defined as vector 79357.png given by the diagonal of the parallelogram OACB, as shown in the figure, i.e., 79363.png.

Scalar product of two vectors

Definition 79369.png = 79375.png cos θ, where θ is the angle between the vectors, where 0 ≤ θπ.
For θ acute, obtuse, and right angle 79383.png is positive, negative, and zero respectively.
Properties of scalar product
  1. Geometrical interpretation of scalar product:
    79389.png = 79395.png
    79401.png = projection length of vector 79407.png in the direction of 79416.png
    Similarly, projection length of vector 79423.png in the direction of 79429.png is 79435.png
  2. 79441.png 79449.png
  3. 79455.png (i.e., commutative)
  4. 79464.png(distributive)
  5. 79470.png (where l and m are scalars)
  6. 79476.png are perpendicular to each other 79482.png
  7. 79490.png = 79496.png
  8. If 79502.png then 79509.png = a1b1 + a2b2 + a3b3
  9. If 79518.png are non-zero, then the angle between them is given by 79528.png
  10. Let 79536.png, Taking dot product with 79542.png and 79548.png alternatively, we have x = 85910.pngThen 79567.png

Vector (or cross) product of two vectors

Definition 79573.png = |a| |b| sin θ 79600.png, where θ is the angle between 79606.png, (0 ≤ θπ), and 79600.png is a unit vector along the line perpendicular to both 79619.png.
Properties of vector product
  1. 79625.png
  2. 79631.png
  3. 79637.png
  4. 79643.png
  5. Two non- zero vectors 79649.png are collinear if and only if 79664.png
  6. If 79671.png, then 79677.png = (a2b3a3b2)79688.png
  7. 79694.png
    = 2 (Area of triangle AOC)
    = Area of parallelogram.
    The area of the triangle OAB is 79707.png.
  8. The unit vectors perpendicular to the plane of 79716.png
  9. If 79728.pngare diagonals of parallelogram; then its
    Area = 79734.png

Scalar triple product

The scalar triple product of three vectors 79740.png is defined as 79746.png
We denote it by 86040.png.
Properties of scalar triple product
  1. 79755.png
    i.e., 86020.png
    = 86028.png
  2. 79761.png
    then 79767.png
    Here 79773.png represents (and is equal to) the volume of the parallelepiped whose adjacent sides are represented by the vectors 79783.png Three vectors 79789.png are coplanar if 86060.png = 0
  3. 86066.png
  4. 86072.png = 79796.png
  5. 79802.png, i.e., position of dot and cross can be interchanged without altering the order. Hence it is also represented by 79808.png
  6. 79816.png (where k is a scalar)
  7. 79822.png i.e., if any two vectors are same then vectors are coplanar
  8. 79828.png
  9. Four points with position vectors 79836.png and 79842.png are coplanar if 86078.png

Vector triple product

The vector triple product of three vectors 79848.png is the vector:
also 79862.png
In general, 79871.png
If 79877.pngthen the vector 79883.pngare collinear.
79889.png is a vector perpendicular to 79895.png but 86095.png is vector perpendicular to the plane of 79901.png.
the vector 79909.png must lie in plane of 79918.png


Also 83249.png
which shows that vector lies in the plane of 83237.png and also in the plane of vectors 83231.png.
Thus the vector lies along the common section of the plane of 83225.pngand the plane of 83219.png.

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