Class 12

Math

Algebra

Vector Algebra

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $a=2i^+3j^ −k^$ and $b=i^−2j^ +k^$.

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Find the direction cosines of the vector $i^+2j^ +3k^˙$

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If $b$ is a vector whose initial point divides thejoin of $5i^and5j^ $ in the ratio $k:1$ and whose terminal point is the origin and $∣∣ b∣∣ ≤37 ,thenk$ lies in the interval a. $[−6,−1/6]$ b. $(−∞,−6]∪[−1/6,∞)$ c. $[0,6]$ d. none of these

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If $a,b,andc$ be three non-coplanar vector and $a_{prime},b_{prime}andc_{′}$ constitute the reciprocal system of vectors, then prove that $r=(ra˙_{′})a+(rb˙_{_{′}})b+(rc˙_{_{′}})c$ $r=(ra˙_{′})a_{′}+(rb˙_{_{′}})b_{′}+(rc˙_{′})c_{′}$

If the resultant of three forces $F_{1}=pi^+3j^ −k^,F_{2}=6i^−k^andF_{3}=−5i^+j^ +2k^$ acting on a parricle has magnitude equal to 5 units, then the value of $p$ is a. $−6$ b. $−4$ c. $2$ d. $4$

Examine the following vector for linear independence: (1) $i+j +k,2i+3j −k,−i−2j +2k$ (2) $3i+j −k,2i−j +7k,7i−j +13k$