Vector Practice Problems

Introduction for vector practice problems:

Vector space model is nothing but an algebraic model, which is normally used for representing text documents as vectors of identifiers. It is used in several purposes such as information filtering, indexing etc. Vectors plays a significant role in the field of mathematics. In this article vector practice problems, we are going to discuss few example and practice problems based on vectors.

Example Problems for 'vector Practice Problems':

Example problem 1:

If the position vectors of A and B are `3 veci - 7 vecj - 7 veck` and `5 veci + 4 vecj + 3 veck` , find `vec(AB)` and determine its magnitude and direction cosines.

Solution:

Let O be the origin. Then

`vec(OA) ` = `3 veci` −` 7 vecj` − `7 veck` ,

`vec(OB)` =` 5 veci` +` 4 vecj` + `3 veck`

` vec(AB)` = `vec(OB)` − `vec(OA)` = (`5veci` + `4vecj ` + `3veck` ) - (`3veci` - `7vecj` - `7veck` )

After simplify this, we get

`vec(AB)` = `2veci` + `11vecj` +` 10veck`

` |vec(AB)|` = `sqrt[(2)^2 + (11)^2 + (10)^2]` = 15

The direction cosines are `2/15` , `11/15` , `10/15`

Example problem 2:

Find the magnitude and direction cosines of `2 veci` −` vecj` + `7 veck`

Solution:

Magnitude of `2veci - vecj + 7veck` = `|2veci - vecj + 7veck|`  = `sqrt[(2)^2 +(-1)^2 + (7)^2]`

After simplify this, we get

=` sqrt(4 + 1 + 49)` = `sqrt(54)` = `3sqrt(6)`

Direction cosines of `2veci - vecj + 7veck` are `2/(3sqrt(6))` , `-1/(3sqrt(6))` , `7/(3sqrt(6))`

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Practice Problems for 'vector Practice Problems':

Practice problem 1:

The position vectors of the vertices's A, B, C of a triangle ABC are respectively `2veci ` + `3 vecj` + `4 veck` , `- veci` + `2 vecj` `- veck` and `3 veci` −` 5 vecj` + `6veck`. Find the vectors determined by the sides and calculate the length of the sides.

Answer:` vec(AB)` = `-3veci` - `vecj` - `5veck` ; `vec(BC)` = `4veci` `- 7vecj` + `7veck` ; `vec(CA)` = `-veci ` + `8vecj` `- 2veck`

AB = √35, BC = √114, CA = √69

Practice problem 2:

If the vectors `veca` = `2 veci` − `3 vecj ` and` vecb ` = − `6vec i ` + `mvec j` are collinear, find the value of m.

Answer: m = 9

Practice problem 3:

If the vertices's of a triangle have position vectors `veci ` + `2 vecj` + `3 veck` , `2veci` + `3 vecj` +` veck` and` 3 veci` + `vecj ` + `2 veck` , find the position vector of its centroid.

Answer: Position vector = 2( `veci` + `vecj ` + `veck` )

Practice problem 4:

Examine whether the vectors` veci` + `3 vecj` + `veck` , `2 veci` − `vecj ` − `veck` and `7 vecj` +` 5 veck` are coplanar.

Answer: non-coplanar vectors