Scalar Triple Product

Question

Three vectors a = i + 2j + 3k, b = 2i + j + k and c = 3i + j + 2k are given; compute their scalar triple product [a b c].

RankGuru · Accepted Answer

The scalar triple product [a b c] equals a . (b x c) and is evaluated as the determinant whose rows are the components of the three vectors. Geometrically it gives the signed volume of the parallelepiped spanned by the vectors, and its vanishing signals coplanarity, a frequently tested JEE Advanced criterion. Setting up the determinant with rows (1,2,3), (2,1,1), (3,1,2), expand along the first row: 1[(1)(2) - (1)(1)] - 2[(2)(2) - (1)(3)] + 3[(2)(1) - (1)(3)] = 1(2 - 1) - 2(4 - 3) + 3(2 - 3) = 1(1) - 2(1) + 3(-1) = 1 - 2 - 3 = -4. Option 6 takes the wrong sign and an inflated magnitude; option 0 would mean the vectors are coplanar, which the nonzero determinant refutes; option -12 triples a minor erroneously. Plausibility check: since the determinant is nonzero, the three vectors are linearly independent and span a genuine three-dimensional parallelepiped, consistent with a nonzero volume.

Scalar Triple Product

Select the correct option:

🔒 Solution Hidden from View

More vector algebra Practice Questions

The scalar triple product [ $a$ $b$ $c$ ] represents

Three vectors $a$ , $b$ , $c$ are coplanar if their scalar triple product is

The shortest distance between two skew lines is computed using which expression involving their dire...

If the scalar triple product [a b c] equals 7 for three given vectors, what is the value of the rear...

Three unit vectors a, b and c satisfy a + b + c = 0; using this constraint, evaluate the sum a . b +...

Two vectors of magnitudes 5 and 12 act at a point with an included angle of 90 degrees; evaluate the...

For two vectors with magnitudes 4 and 3 enclosing a 30 degree angle, evaluate the sum of squares |a ...

For what value of the scalar lambda are the three vectors i + j + k, i + lambda j + 2k and 2i + j + ...

Two vectors a and b have magnitudes 3 and 5 respectively while their dot product equals 15/2; find t...

A vector makes equal acute angles with all three coordinate axes; determine the value of each of its...

Find the scalar value m so that the vector 2i + m j + k becomes perpendicular to the vector i - 3j +...

Determine the value of the scalar t for which the vectors i + 2j, 2i + tj and the resulting set beco...

About This Question

Solution

Scalar Triple Product

Select the correct option:

🔒 Solution Hidden from View

More vector algebra Practice Questions

The scalar triple product [a b c] represents

Three vectors a, b, c are coplanar if their scalar triple product is

The shortest distance between two skew lines is computed using which expression involving their dire...

If the scalar triple product [a b c] equals 7 for three given vectors, what is the value of the rear...

Three unit vectors a, b and c satisfy a + b + c = 0; using this constraint, evaluate the sum a . b +...

Two vectors of magnitudes 5 and 12 act at a point with an included angle of 90 degrees; evaluate the...

For two vectors with magnitudes 4 and 3 enclosing a 30 degree angle, evaluate the sum of squares |a ...

For what value of the scalar lambda are the three vectors i + j + k, i + lambda j + 2k and 2i + j + ...

Two vectors a and b have magnitudes 3 and 5 respectively while their dot product equals 15/2; find t...

A vector makes equal acute angles with all three coordinate axes; determine the value of each of its...

Find the scalar value m so that the vector 2i + m j + k becomes perpendicular to the vector i - 3j +...

Determine the value of the scalar t for which the vectors i + 2j, 2i + tj and the resulting set beco...

About This Question

Solution

The scalar triple product [ $a$ $b$ $c$ ] represents

Three vectors $a$ , $b$ , $c$ are coplanar if their scalar triple product is