(1) Verify whether the following ratios are direction cosines of some vector or not.
(i) 1/5 , 3/5 , 4/5
(ii) 1/√2, 1/2 , 1/2
(iii) 4/3, 0, 3/4 Solution
(2) Find the direction cosines of a vector whose direction ratios are
(i) 1 , 2 , 3 (ii) 3 , - 1 , 3 (iii) 0 , 0 , 7 Solution
(3) Find the direction cosines and direction ratios of the following vectors.
(i) 3i vector - 4j vector + 8k vector
(ii) 3i vector + j vector + k vector
(iii) j vector
(iv) 5i vector - 3j vector - 48k vector
(v) 3i vector + 4j vector - 3k vector
(vi) i vector - k vector Solution
(4) A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians. Solution
(5) If 1/2, 1/√2, a are the direction cosines of some vector, then find a. Solution
(6) If (a , a + b , a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c. Solution
(7) Show that the vectors 2i − j + k, 3i − 4j − 4k, i − 3j − 5k form a right angled triangle. Solution
(8) Find the value of λ for which the vectors a = 3i + 2j + 9k and b = i + λj + 3k are parallel. Solution
(9) Show that the following vectors are coplanar
(i) i− 2 j + 3k, − 2i + 3j − 4k, − j + 2k
(ii) 5i + 6 j + 7k, 7i −8 j + 9k, 3i + 20j + 5k Solution
(10) Show that the points whose position vectors 4i + 5j + k, − j − k, 3i + 9j + 4k and −4i + 4j + 4k are coplanar. Solution
(11) If
find the magnitude and direction cosines of
(i) a vector + b vector + c vector
(ii) 3a vector - 2b vector + 5c vector Solution
(12) The position vectors of the vertices of a triangle are i+2j +3k; 3i − 4j + 5k and − 2i+ 3j − 7k . Find the perimeter of the triangle (Given in vectors) Solution
(13) Find the unit vector parallel to 3a − 2b + 4c if a = 3i − j − 4k, b = −2i + 4j − 3k, and c = i + 2 j − k Solution
(14) The position vectors a vector, b vector, c vector of three points satisfy the relation 2a vector - 7b vector + 5c vector. Are these points collinear? Solution
(15) The position vectors of the points P, Q, R, S are i + j + k, 2i+ 5j, 3i + 2j − 3k, and i − 6j − k respectively. Prove that the line PQ and RS are parallel Solution
(16) Find the value or values of m for which m (i + j + k) is a unit vector. Solution
(17) Show that the points A (1, 1, 1), B(1, 2, 3) and C(2, - 1, 1) are vertices of an isosceles triangle Solution
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