Dot Product: JEE Questions & Solutions

Let AB = 2i + 4j - 5k and AD = i + 2j + λ k, λ ∈ ℝ. Let the projection of the vector v = i + j + k on the diagonal AC of the parallelogram…

Let a vector a=√2 i- j+λ k, λ>0, make an obtuse angle with the vector b=-λ² i+4√2 j+4√2 k and an angle θ, π/6<θ<π/2, with the positive…

Let a=2i-j-k, b=i+3j-k and c=2i+j+3k. Let v be the vector in the plane of a and b, such that the length of its projection on the vector c…

For three unit vectors a, b, c satisfying | a- b|² + | b- c|² + | c- a|² = 9 and |2 a + k b + k c| = 3, the positive value of k is:

Let a and b be two unit vectors such that the angle between them is π/3. If λ a + 2 b and 3 a - λ b are perpendicular to each other, then…

If a is a non-zero vector such that its projections on the vectors 2i - j + 2k, i + 2j - 2k, and k are equal, then a unit vector along a is:

Consider two vectors u = 3i - j and v = 2i + j - λ k, λ > 0. The angle between them is given by cos⁻¹ ( √5/2√7). Let v = v₁ + v₂, where v₁…

Let the angle θ, 0 < θ < π/2 between two unit vectors a and b be sin⁻¹ ( √65/9). If the vector c = 3a + 6b + 9(a × b), then the value of…

Least positive integral value of α, for which the angle between vectors α i-2j+2k and α i+2α j-2k is acute, is:

The least positive integral value of α for which the angle between the vectors α i - 2j + 2k and α i + 2α j - 2k is acute is:

Let a unit vector u = xi + yj + zk make angles π/2, π/3, 2π/3 with the vectors p₁ = 1/√2i + 1/√2k, p₂ = 1/√2j + 1/√2k, and p₃ = 1/√2i +…

If λ > 0, and θ is the angle between vectors a = i + λ j - 3 k and b = 3 i - j + 2 k, such that a + b and a - b are mutually perpendicular,…

Let a = i + 2j + 3k, b = 2i + 3j - 5k, and c = 3i - j + λ k be three vectors. Let r be a unit vector along b + c. If r · a = 3, then 3λ is…

Vectors a = ai + bj + k and b = 2i - 3j + 4k are perpendicular to each other when 3a + 2b = 7. The ratio of a to b is x/2. The value of x…

If two vectors P = i + 2mj + mk and Q = 4i - 2j + mk are perpendicular to each other, then the value of m will be:

Let α = 4i + 3j + 5k and β = i + 2j - 4k. Let β₁ be parallel to α and β₂ be perpendicular to α. If β = β₁ + β₂, then the value of 5β₂ · (i…

If the vectors a = λ i + μ j + 4k, b = -2i + 4j - 2k, and c = 2i + 3j + k are coplanar, and the projection of a on vector b is √54 units,…

Let a and b be two vectors. Let |a| = 1, |b| = 4 and a · b = 2. If c = (2a × b) - 3b, then the value of b · c is

Let a = 2i + j + k, and b and c be two nonzero vectors such that | a + b + c | = | a + b - c | and b · c = 0. Consider the following two…