**Practice Problems in Vector With Solution :**

Here we are going to see some practice problems in vector. You may also find solution for each problems.

(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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